APPLICATIONS of CUBICAL ARRAYS in the STUDY of FINITE SEMIFIELDS by KELLY CASIMIR AMAN Presented to the Faculty of the Graduate

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APPLICATIONS OF CUBICAL ARRAYS IN THE

STUDY OF FINITE SEMIFIELDS

KELLY CASIMIR AMAN

Presented to the Faculty of the Graduate School of

The University of Texas at Arlington in Partial Fulﬁllment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT ARLINGTON

I owe thanks to many people for helping me reach this milestone in my life, but

I will limit this section to those who have most directly aﬀected the work presented here.

First, I want to thank those who helped with the completion of this dissertation and its defense. Thanks to my advisor, Dr. Minerva Cordero, for her guidance, encouragement, and assistance, as well as allowing me to pursue my interests when conducting the research presented here. I also want to thank the other members of my committee: Dr. Epperson, Dr. Gornet, Dr. Jorgensen, and Dr. Vancliﬀ, for reviewing this work, attending the defense, and providing useful feedback. Lastly, I need to thank Dr. Wene for pointing out some properties of cubical arrays which began the research which led to this dissertation.

I feel it is important to thank the teachers who have most inﬂuenced my work as a mathematician. In addition to Dr. Cordero, Dr. Gornet, and Dr. Vancliﬀ, I must also thank Dr. Krueger and David Smith for instilling in me the goal of making my work as rigorous as possible.

Finally, I want to thank my family and friends for their support. In particular,

I want to thank my wife, Jackie, for changing her schedule to best suit my needs.

June 23, 2014

iii Abstract

APPLICATIONS OF CUBICAL ARRAYS IN THE

STUDY OF FINITE SEMIFIELDS

Kelly Casimir Aman, Ph.D.

The University of Texas at Arlington, 2014

Supervising Professor: Minerva Cordero

It is well known that any ﬁnite semiﬁeld, 푆, can be viewed as an 푛-dimensional

vector space over a finite field or prime order, 픽푝, and that the multiplication in 푆 defines and can be defined by an 푛×푛×푛 cubical array of scalars, 퐴. For any element

푎 ∈ 푆, the matrix, 퐿푎, corresponding to left multiplication by 푎 can be determined from 퐴. In this paper we show that there exists a unique monic polynomial of minimal

degree, 푓 ∈ 픽푝[푥], such that 푓(푎) = 0, and which divides the minimal polynomial of

퐿푎. Furthermore, we show that some properties of 푓 in 픽푝[푥] correspond to properties of 푎 in 푆. These results, in turn, help optimize a method we introduce which uses 퐴

to determine the automorphism group of 푆. We show that under certain conditions

퐴 can be inflated to define a new semifield, 푆[푚], over the field 픽푝푚 , and that inflation preserves isotopism and isomorphism between inflated semifields. Finally, we apply

our results to the 16-element semiﬁelds, and give algebraic constructions for each of

these semiﬁelds for which no construction currently exists.

iv Table of Contents

Acknowledgments ...... iii

Abstract ...... iv

Chapter Page

1. Introduction ...... 1

2. Semiﬁelds and Their Planes ...... 4

2.1 Semiﬁeld Basics ...... 4

2.2 Semiﬁelds as Vector Spaces ...... 6

2.3 Examples ...... 9

2.4 Finite Projective Planes ...... 10

2.5 Isotopism ...... 12

2.6 The Geometric Signiﬁcance of Isotopisms ...... 17

2.7 Autotopisms and Automorphisms ...... 20

3. Cubical Arrays ...... 22

3.1 Cubical Array Basics ...... 22

3.2 Examples ...... 27

3.3 The Triple Product ...... 29

3.4 Isotopism ...... 32

4. The Relationship Between 푆 and 픽푝[푥] ...... 36 4.1 Exponents, Primitive Elements, and Cyclic Bases ...... 37

4.2 Left Order and Minimal Polynomial ...... 38

4.3 Determining the Left Minimal Polynomial ...... 44

4.4 Consequences of the Left Minimal Polynomial ...... 45

v 5. Determining Aut(푆) ...... 49

5.1 Investigating 퐾(푆) ...... 49 5.2 Utilizing Properties of Automorphisms ...... 52

5.3 Automorphisms of System V ...... 53

5.4 Automorphisms of Sandler’s Construction of order 332 ...... 55 6. Constructions and Enumerations ...... 58

6.1 Early Enumeration - Kleinfeld and Walker ...... 58

6.2 Recent Enumerations by Rúa, Combarro, and Ranilla ...... 59

6.3 Reverse Decimal Matrix Notation ...... 61

6.4 Examples ...... 64

6.5 Knuth Cubes and Constructions ...... 65

6.6 Albert Binary Semiﬁelds ...... 65

6.7 Knuth Binary Semiﬁelds ...... 67

6.8 Sandler’s Construction ...... 69

6.9 Petit’s Construction ...... 70

6.10 Dickson’s Even-Dimensional Semiﬁelds ...... 71

6.11 Quadratic Over a Weak Nucleus ...... 72

6.12 Albert’s Twisted Fields ...... 74

7. Extending a Subﬁeld of a Semiﬁeld ...... 76

7.1 Cubical Arrays Over Extension of 픽푝 ...... 76 7.2 Determining Valid Extension Fields ...... 79

7.3 Results of Extension ...... 81

8. Determining Constructions of the Semiﬁelds of Order 16 ...... 83

8.1 Verifying Kleinfeld’s Results ...... 83

8.2 Eliminating Known Constructions ...... 85

8.3 Determining Constructions Part 1: Quadratic Over 픽4 ...... 86 vi 8.4 Determining Constructions Part 2: Almost Quadratic Over 픽4 . . . . 92

8.5 Determining Constructions Part 3: Not Quadratic Over 픽4 ...... 94 8.6 A Catalog Of The 16-Element Semiﬁelds ...... 95

References ...... 121

Biographical Statement ...... 124

vii Chapter 1

Introduction

Finite semifields satisfy all of the axioms for finite fields except that their multiplication is not assumed to be associative or commutative. In this paper we study the algebraic properties of finite semifields through the use of cubical arrays. One motivation for semifield research is its applications to the theory of finite geometry, particularly finite translation planes. However, we take an algebraic approach to the work presented here. Nevertheless, we include a brief description of the geometric motivations for the concepts of isotopism and the dual of a semifield.

Chapters 2 and 3 provide background information for our results. They are based on the work of Donald Knuth in [11]. In chapter 2, we provide the formal definition of a finite semifield, and prove that any finite semifield, 푆, is an 푛-dimensional vector space over a prime order finite field, 픽푝. Furthermore, we show that left or right multiplication by each element in 푆 has a corresponding linear transformation. We also describe the relationship between finite semifields and finite projective planes, and introduce the geometrically and algebraically important concept of an isotopism between two semifields. In chapter 3, we show that the multiplication in a semifield defines an 푛 × 푛 × 푛 array of scalars known as a cubical array, 퐴, and show the necessary conditions for a cubical array to define a semifield. We will call such cubical arrays Knuth cubes. We show how, for any 푎 ∈ 푆, the matrix corresponding to left multiplication by 푎, 퐿푎 can be determined from 퐴, and show how cubical arrays of isotopic semifields are related.

The remainder of this work is devoted to the results of our research. In chapter

1 4, we investigate the action of polynomials on elements of ﬁnite semiﬁelds. Due to the

fact that semiﬁeld multiplication is nonassociative, we adopt the convention of using

two exponential notations to refer to repeated left or right multiplication. Doing so

allows us to prove that there exists a unique left minimal polynomial, 푓 ∈ 픽푝[푥] such that, when the exponents of 푓 are used for left multiplication, 푓(푎) = 0, and proves a

number of results regarding the relationship between 푎, 푓, and 퐿푎. Analogous results hold if right multiplication is used in place of left multiplication.

In chapter 5, we describe a subgroup, 퐶(푛, 푝), of 퐺퐿(푛, 푝), and show how

퐶(푛, 푝) can be used to generate the set of all Knuth cubes which deﬁne 푆, 퐾(푆).

We show necessary and suﬃcient conditions for matrices in 퐶(푛, 푝) to correspond to automorphisms of 푆. This allows the automorphism group of 푆 to be determined by testing each matrix in 퐶(푛, 푝), and we use the results of chapter 4 to greatly reduce the number of matrices which need to be tested.

In chapter 6, we outline a number of classical algebraic constructions for finite semifields and describes a Knuth cube generated by each construction. Using these results, it is possible to look at the Knuth cubes in 퐾(푆) and see if any is of the proper form for each construction. This provides a method by which cubical arrays can be used to identify whether a particular construction will generate a particular semifield. The primary use of these results is in chapter 8 to identify which of the

16-element semiﬁelds can be generated by existing constructions.

The results in chapter 7 were inspired by the work of chapter 6, and follow directly from the results in chapters 2 and 3. We provide necessary and sufficient conditions under which 퐴 can define a semifield over 픽푝푚 , denoted 푆[푚]. We then ′ prove that if 푆[푚] exists, then 푆 and 푆 are isotopic semifields if and only if 푆[푚] and ′ 푆[푚] are isotopic semifields. In chapter 8, we applyin our results to the 16-element semifields. Currently al- 2 gebraic constructions exist for representatives of the three isotopism classes for these semifields, and we show how the known Knuth cubes can be used to provide algebraic constructions for each of the twenty-four isomorphism classes of these semifields. We conclude with a list of representative for each isomorphism class of the 16-element semifields, and for each representative we provide a Knuth cube, an algebraic construction, a list of automorphisms defined with respect to the algebraic construction, and a list of the elements contained in the most commonly studied semifield subsets.

3 Chapter 2

Semiﬁelds and Their Planes

The term semifield, and the definition which follows, were first introduced by

Knuth in [11]. Even now, although the term has been widely adopted, these systems are sometimes referred to as “nonassociative division rings” or “distributive quasi-

fields”. In this chapter we will define what semifields are, describe their general form, and look at the geometries which inspired them. All of the results and proofs in this chapter are taken from [11], and are included here for the sake of completeness. Some of the notation has been altered to help avoid ambiguity and to suit the needs of the results in the later chapters. The proofs have been slightly altered as well to be more rigorous.

2.1 Semiﬁeld Basics

Definition 2.1. A finite semifield is a finite set, 푆, together with two binary operations, denoted + and ∗, which satisfies the following axioms, for all 푎, 푏, 푐 ∈ 푆:

1. (푆, +) forms a group, with identity element 0.

2. If 푎 ∗ 푏 = 0, then 푎 = 0 or 푏 = 0.

3. 푎 ∗ (푏 + 푐) = 푎 ∗ 푏 + 푎 ∗ 푐 and (푎 + 푏) ∗ 푐 = 푎 ∗ 푐 + 푏 ∗ 푐.

4. There exists 푒 ∈ 푆 such that 푎 ∗ 푒 = 푒 ∗ 푎 = 푎. In the following work we will use the term semifield to refer to a finite semifield, and, unless otherwise noted, all other sets and systems mentioned will be assumed

finite. Notice that any finite field will satisfy this definition, and thus every finite

field is a semifield. The key difference is that semifield multiplication is not assumed

4 to be associative or commutative. By Wedderburn’s theorem, associativity will imply

commutativity, but the converse is not true. As noted in [7], there exist commutative

semiﬁelds which are not associative, so the converse is not true.

Many of the results in this work could therefore be applied to the study of ﬁnite

fields, although there is little need since finite fields are already well understood.

Because of this, we will sometimes include examples of results applied to ﬁelds in

addition to semiﬁelds to highlight diﬀerences between the systems. And, when it is

necessary to emphasize that a semiﬁeld is not a ﬁeld, we will refer to it as a proper

semiﬁeld. In other words, in a proper semiﬁeld 푆, there exist 푎, 푏, 푐 ∈ 푆 such that

푎 ∗ (푏 ∗ 푐) ≠ (푎 ∗ 푏) ∗ 푐. Associative multiplication is such a common and fundamental assumption in mathematics that the eﬀects of its absence are not always obvious. For example, exponential notation for multiplication is not well deﬁned for powers greater than 2.

The term 푥2 is unambiguous and stands for 푥 ∗ 푥, but 푥3 could mean 푥 ∗ (푥2) or

(푥2) ∗ 푥. A similar issue arises with the idea of multiplicative inverses. Consider the following well known lemma.

Lemma 2.2 (Knuth, [11]). Let 푆 be a semiﬁeld, and let 푎, 푏 ∈ 푆 be nonzero. Then there exist unique 푥 and 푦 in 푆 such that 푎 ∗ 푥 = 푏 and 푦 ∗ 푎 = 푏.

Proof. Let 푆 = {0, 푐1, 푐2, ..., 푐푞}, and consider the set {푎 ∗ 푐1, 푎 ∗ 푐2, ..., 푎 ∗ 푐푞}. Since

푆 can have no zero divisors, 푎푐푖 ≠ 0 for all 푖. Suppose there exist 푖, 푗 such that

푎 ∗ 푐푖 = 푎 ∗ 푐푗. Then 푎 ∗ (푐푖 − 푐푗) = 0, and 푐푖 = 푐푗. Thus each 푎 ∗ 푐푖 must be distinct,

and there must exist a unique 푖 such that 푎푐푖 = 푏, thus 푥 = 푐푖. A similar proof shows that there exists a unique 푦 such that 푦 ∗ 푎 = 푏.

Consider a nonzero element 푎 ∈ 푆. This lemma shows that there exist 푥, 푦 ∈ 푆

such that 푎 ∗ 푥 = 푒 and 푦 ∗ 푎 = 푒. For this reason a finite semifield, like a finite field,

5 will have no nontrivial ideals. There are still subsets of 푆 which are of particular interest.

Definition 2.3. Let 푆 be a finite semifield. Then the left nucleus, middle nucleus, and right nucleus are the subsets 푁푙, 푁푚, and 푁푟 of 푆 defined by

(2.1) 푁푙 = {푎 ∈ 푆|푎 ∗ (푥 ∗ 푦) = (푎 ∗ 푥) ∗ 푦 ∀푥, 푦 ∈ 푆}

(2.2) 푁푚= {푎 ∈ 푆|푥 ∗ (푎 ∗ 푦) = (푥 ∗ 푎) ∗ 푦 ∀푥, 푦 ∈ 푆}

(2.3) 푁푟 = {푎 ∈ 푆|푥 ∗ (푦 ∗ 푎) = (푥 ∗ 푦) ∗ 푎 ∀푥, 푦 ∈ 푆}

The nucleus, 푁, of 푆 is the intersection of 푁푙, 푁푚, and 푁푟. The left, middle, and right nucleus of a semiﬁeld are sometimes referred to collectively as the nuclei of 푆. This terminology is still not entirely standard, and some authors may use the term nucleus to refer to 푁푙 when it is the only nucleus being considered.

Definition 2.4. Let 푆 be a semifield. The center of 푆, denoted 푍, is the subset of 푆 defined by

(2.4) 푍 = {푥 ∈ 푆|푥 ∗ 푦 = 푦 ∗ 푥 ∀푦 ∈ 푆}

It is worth noting that 푁푙, 푁푚, 푁푟, and 푁 are all isomorphic to finite fields, by Wedderburn’s theorem. Furthermore, 푍 ∩ 푁 is also isomorphic to a finite field, and is sometimes referred to as the associative center of 푆. But 푍 may not be isomorphic to a finite field, and there exist cases where 푍 = 푆 and 푆 is a proper semifield.

2.2 Semiﬁelds as Vector Spaces

Another useful consequence of Lemma 2.2 is that 푎 ∗ 0 = 0 ∗ 푎 = 0 for all 푎 ∈ 푆, since 푎 ∗ 푏 = 0 implies that 푎 or 푏 is zero. This, coupled with the fact that any semiﬁeld is a group under addition, yields the following useful result. 6 Lemma 2.5 (Knuth, [11]). Let 푆 be a semiﬁeld. Then the following are true:

1. The group (푆, +) is Abelian.

2. (푆, +) has characteristic 푝, where 푝 is a prime.

3. 푆 is an 푛-dimensional vector space over the ﬁnite ﬁeld of order 푝, 픽푝. 4. |푆| = 푝푛 for a prime 푝 and positive integer 푛.

Proof. Let 푎, 푏 ∈ 푆 and let 푒 denote the multiplicative identity of 푆. By distributivity, we have

(푎 ∗ 푒 + 푎 ∗ 푒) + (푏 ∗ 푒 + 푏 ∗ 푒) = (푎 + 푏) ∗ (푒 + 푒) = (푎 ∗ 푒 + 푏 ∗ 푒) + (푎 ∗ 푒 + 푏 ∗ 푒)

Since (푆, +) is a group, we then have

(푎 ∗ 푒 + 푎 ∗ 푒) + (푏 ∗ 푒 + 푏 ∗ 푒) = (푎 ∗ 푒 + 푏 ∗ 푒) + (푎 ∗ 푒 + 푏 ∗ 푒)

= (푎 + 푎) + (푏 + 푏) = (푎 + 푏) + (푎 + 푏)

= 푎 + (푎 + 푏) + 푏 = 푎 + (푏 + 푎) + 푏

= 푎 + 푏 = 푏 + 푎

Thus (푆, +) is Abelian. Now let 푎 ≠ 0. For any 푘 ∈ ℤ+, let (푘푎) denote 푎 added to itself 푘 times. Let 푝 denote the smallest integer such that (푝푎) = 0 and suppose there exist 푚, 푛 ∈ ℤ+ such that 푛푚 = 푝. This gives us

(푚푎) ∗ (푛푎) = (푚(푛(푎 ∗ 푎))) = ((푚푛)(푎 ∗ 푎)) = ((푚푛)푎) ∗ 푎 = (푝푎) ∗ 푎 = 0 ∗ 푎 = 0

Then (푚푎) = 0 or (푛푎) = 0, which implies 푚 = 푝 or 푛 = 푝. Thus 푝 is prime. Now let 푎, 푏 ∈ 푆 and let 푝1 and 푝2 denote the additive orders of 푎 and 푏 respectively, and let 푥 denote the multiplicative order of (푎 + 푏). From the previous argument, 푥 must be prime, but since 푝1 and 푝2 must both divide 푥, the only way this can be true is if 푝1 = 푝2 = 푥. Thus all elements of 푆 have the same additive order. Thus 푆 must be a vector space over the ﬁnite ﬁeld of 푝 elements, 픽푝, with scalar multiplication 7 referring to repeated addition. Thus we have |푆| = 푝푛, where 푛 is the dimension of

푆 over 픽푝.

Definition 2.6. Let 푆 be a semifield where |푆| = 푝푛. Then the order of 푆 is 푝푛. When discussing the number of elements in a semifield we will sometimes specify

that 푆 has 푝푛 elements, is a 푝푛-element semifield, or has order 푝푛. Since finite fields and vectors over finite fields are also used throughout this work, we will use the

푛 notation 픽푝 to denote the set of 푛-dimensional vectors over 픽푝, and 픽푝푛 to denote the finite field of 푝푛 elements. The fact that semifields are vector spaces provides some useful properties. If

푆 and 푆′ are semiﬁelds of order 푝푛, then (푆, +) and (푆′, +) are both isomorphic to

푛 (픽푝 , +). Consider elements 푎, 푏 ∈ 푆. Because multiplication is distributive, there

exist linear transformations 퐿푎 and 푅푏 such that

(2.5) 푎 ∗ 푏 = 퐿푎(푏) = 푅푏(푎)

By Lemma 2.2, these transformations must be bijective. This is a very useful property,

since it allows us to view multiplication in terms of linear transformations. For

−1 −1 instance, if 푅푏(푎) = 푎 ∗ 푏, then there exists 푅푏 such that 푅푏 (푎 ∗ 푏) = 푎. Some care should be taken when working with these functions though. Consider the diﬀerences

between the following: 2 푅푏 (푎) = (푎 ∗ 푏) ∗ 푏

푅푏2 (푎) = 푎 ∗ (푏 ∗ 푏)

The ﬁrst function is 푅푏(푅푏(푎)) which is distinct from 푅푏2 (푎), and clearly these two functions are distinct.

Finally, we can let 1 denote the multiplicative identity of 푆 without fear of

ambiguity. Let 푖, 푗 ∈ 픽푝. Then (푖푗)푒 = (푖푒)(푗푒) and (푖 + 푗)푒 = 푖푒 + 푗푒. Thus there is a subﬁeld of 푆 isomorphic to 픽푝, and the multiplicative identity of 푆 is isomorphic to 8 the multiplicative identity of 픽푝. We will continue to let 푒 denote the multiplicative identity of 푆 for now to keep the notation as clear as possible, but we will switch to

1 in later chapters when it is more convenient.

2.3 Examples

As previously mentioned, every finite field is also a semifield. The following

result gives a necessary condition for a semiﬁeld to be proper.

Lemma 2.7 (Knuth, [11]). If 푆 is a proper semiﬁeld of order 푝푛, then 푛 ≥ 3.

2 Proof. If |푆| = 푝, then 푆 must be isomorphic to 픽푝. If |푆| = 푝 , then 푆 is two-

dimensional over 픽푝, and has a basis of the form {푒, 푥}. The multiplication in 푆 is 2 therefore determined by 푥 ∗ 푥 = 푎푥 + 푏푒 for 푎, 푏 ∈ 픽푝. Now consider 푥 − 푎푥 − 푏푒 = 0. This cannot be factored, otherwise 푆 would contain zero divisors. Thus it is irreducible and 푆 ≅ 픽푝2 .

Knuth also proves in section 6.1 of [11] that if |푆| = 8, then 푆 ≅ 픽8. Thus the smallest proper semiﬁelds are of order 16. Throughout this work we will use the

following three 16-element semiﬁelds for examples.

Each of these examples has elements of the form 푎 + 휆푏, where 푎, 푏 ∈ 픽4 and 2 픽4 = {0, 1, 휔, 휔 = 1 + 휔}. Addition is defined in the usual way, with (푎 + 휆푏) + (푐 + 휆푑) = (푎 + 푐) + 휆(푏 + 푑), so the only distinguishing feature of these semifields is how multiplication is defined. The names “system V” and “system W” were used by

Knuth in [11] as a means of identifying these constructions throughout his work, and

9 we will do the same here.

(2.6) 픽16 (푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏푑휔) + 휆(푏푐 + 푎푑 + 푏푑)

(2.7) System W (푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏2푑휔) + 휆(푏푐 + 푎2푑)

(2.8) System V (푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏2푑) + 휆(푏푐 + 푎2푑 + 푏2푑2)

Notice that the multiplication in both system V and system W is very similar to

that in 픽16, yet has been modiﬁed in a way which makes it nonassociative. Also,

each of these three semiﬁelds can be viewed as a right vector space over 픽4, or as a

vector space over 픽2. A valid basis over 픽2 for all three of these semiﬁelds would be {1, 휔, 휆, 휆휔}, and, by distributivity, the multiplication in each semiﬁeld is determined by the products of the elements of this basis. This is the motivation behind the results in chapter 3.

2.4 Finite Projective Planes

An important application of semifields is in the study of finite projective planes, and a large portion of the existing research on semifields relates to this application.

Although the later results in this work were inspired by a purely algebraic motivation, the results in this section will provide a useful tool for classifying semiﬁelds.

Definition 2.8. A finite semifield plane is a finite set, 휋, whose elements are called points, together with a collection of subsets, called lines, which is said to be coordi-

natized by a semiﬁeld, 푆, if the points and lines of 휋 can be expressed in the following

way, for 푎, 푏 ∈ 푆:

⎧ ⎧ { (1, 푎, 푏) { [1, 푎, 푏] { { points = lines = ⎨ (0, 1, 푎) ⎨ [0, 1, 푎] { { { { ⎩ (0, 0, 1) ⎩ [0, 0, 1] 10 where a point (푥1, 푥2, 푥3) is contained in a line [푦1, 푦2, 푦3] if and only if

(2.9) 푦1푥3 = 푥2 ∗ 푦2 + 푥1푦3

The values 0 and 1 used in the deﬁnition will be treated as scalar multiples of the semiﬁeld elements. The general term for the relationship between the points and lines in a plane is incidence, and we will refer to equation 2.9 as the incidence equation.

This notation for points and lines is not standard either; Knuth used it in [11] and it is based on a notation used by Albert in [3]. While it does not provide an intuitive view of the geometry of a semiﬁeld plane, it does give all of the points a similar form, and provides a simple means of testing whether a point is on a line via equation 2.9.

A finite semifield plane is a special case of the more general definition of a finite projective plane. We include this definition in order to better illustrate the geometric nature of the semifield planes.

Definition 2.9. A finite projective plane is a finite set, 휋, whose elements are called points, together with a collection of subsets, called lines, which satisfy the following axioms:

1. Two distinct points are contained in one and only one line.

2. The intersection of two distinct lines contains one and only one point.

3. There exist four points, no three of which are contained in the same line.

Standard geometric terms are commonly used when discussing projective planes. If a point is contained on a line it is said to lie on that line, and the line is said to pass through the point.

A more standard approach to coordinatizing projective planes using the more general ternary rings can be found in the early chapters of “Aﬃne Planes with Tran- 11 sitive Collineation Groups” by Kallaher (see [9]). It can be proven that a semiﬁeld

plane satisﬁes the axioms for a projective plane, and it is easily seen by the deﬁnition

that a semifield plane contains 푞2 + 푞 + 1 points and 푞2 + 푞 + 1 lines, where 푞 is the order of the semifield. This property is true for finite projective planes as well, where

푞 is the number of elements in the ternary ring used to coordinatize the plane. The following notation will become useful when discussing projective planes.

Given points (푎1, 푎2, 푎3) and (푏1, 푏2, 푏3), the line passing through them will be denoted (푎1, 푎2, 푎3) ∶ (푏1, 푏2, 푏3). Given lines [푦1, 푦2, 푦3] and [푧1, 푧2, 푧3], the point where they intersect will be denoted [푦1, 푦2, 푦3] ∩ [푧1, 푧2, 푧3]. Using the incidence equation, we can then derive the following deﬁnitions for the points and lines of a plane:

[0, 0, 1] = (0, 1, 0) ∶ (0, 0, 1) (0, 0, 1) = [0, 1, 0] ∩ [0, 0, 1]

(2.10) [0, 1, 푎] = (0, 0, 1) ∶ (1, −푎, 푏) (0, 1, 푎) = [0, 0, 1] ∩ [1, 푎, 푏]

[1, 푎, 푏] = (0, 1, 푎) ∶ (1, 0, 푏) (1, 푎, 푏) = [0, 1, −푎] ∩ [1, 0, 푏]

Some deﬁnitions, such as [0, 1, 푎] = (0, 0, 1) ∶ (1, −푎, 푏) may seem ambiguous. Through two distinct points there exists a unique line, so how could [0, 1, 푎] stay constant for any choice of 푏? The incidence equation veriﬁes this, but a more intuitive answer is that [0, 1, 푎] corresponds to the equation 푥 = −푎, which is a vertical line. Thus

[0, 1, 푎] = (0, 0, 1) ∶ (1, −푎, 푏) for any choice of 푏.

2.5 Isotopism

In this section we discuss isotopisms, which can be thought of as a generalization of the concept of isomorphisms. Unlike isomorphisms, isotopisms do not preserve any of the multiplicative properties of a semifield except for nonassociativity. Thus is is possible to have commutative semifields isotopic to noncommutative semifields,

12 and to even have semiﬁelds isotopic to systems which do not posses a multiplicative

identity.

Deﬁnition 2.10. Let 푆 and 푆′ be semiﬁelds of order 푝푛, with multiplication ∗ and ⋆

′ respectively. Then 푆 is said to be isotopic to 푆 if there exist nonsingular 픽푝-linear transformations, 퐹 , 퐺, and 퐻, from 푆 to 푆′ such that, for all 푎, 푏 ∈ 푆,

(2.11) 퐻(푎 ∗ 푏) = 퐹 (푎) ⋆ 퐺(푏)

The ordered set {퐹 , 퐺, 퐻} is said to be an isotopism from 푆 to 푆′. There are a number of important things to note regarding this deﬁnition. First, we could have used a more general deﬁnition of isotopism using ternary rings which assumes the functions are bijective but does not assume linearity, and many of the following results could be proven in the more general case. In particular, Theorem

2.13 can be proven without assuming linearity (see Theorem 3.2.3 in [11]), but shows that linearity is sufficient for finding all semifields isotopic to a given semifield. Thus nothing is lost by assuming linearity, and it will make the following results more straightforward.

Second, if 퐹 = 퐺 = 퐻, then an isotopism is an isomorphism. Also, if 푆 and 푆′

are isotopic, with isotopism {퐹 , 퐺, 퐻} from 푆 to 푆′, then there is also an isotopism

{퐹 −1, 퐺−1, 퐻−1} from 푆′ to 푆. In some cases is is more convenient to deﬁne the

isotopism going from 푆′ to 푆, so it is worth noting that this notion is well deﬁned.

Finally, if there are semiﬁelds 푆, 푆1, and 푆2 such that 푆 is isotopic to 푆1, with isotopism {퐹1, 퐺1, 퐻1}, and 푆1 is isotopic to 푆2, with isotopism {퐹2, 퐺2, 퐻2}, then

푆 is isotopic to 푆2 by the isotopism {퐹2 ∘ 퐹1, 퐺2 ∘ 퐺1, 퐻2 ∘ 퐻1}. Lemma 2.11 (Knuth, [11]). Let {퐹 , 퐺, 퐻} be an isotopism from (푆, +, ∗) to

(푆′, +, ⋆). Then

(2.12) 퐻 = 푅 ∘ 퐹 = 퐿 ∘ 퐺 13 where 퐿, 푅 ∶ 푆′ ↦ 푆 are deﬁned by 퐿(푥) = 퐹 (푒) ⋆ 푥, 푅(푥) = 푥 ⋆ 퐺(푒) and 푒 is the

multiplicative identity of 푆.

Proof. Let 푎 ∈ 푆. By the deﬁnition of isotopism we have

퐻(푎 ∗ 푒) = 퐹 (푎) ⋆ 퐺(푒)

= 퐻(푎) = 푅(퐹 (푎))

퐻(푒 ∗ 푎) = 퐹 (푒) ⋆ 퐺(푎)

= 퐻(푎) = 퐿(퐺(푎))

Theorem 2.12 (Knuth, [11]). Let 푆 be a semiﬁeld of order 푝푛. Then the number of nonisomorphic semiﬁelds which are isotopic to 푆 is at most (푝푛 − 1)2.

Proof. Let (푆1, +, ⋆1) and (푆2, +, ⋆2) be semiﬁelds isotopic to a semiﬁeld 푆, and let

{퐹1, 퐺1, 퐻1} and {퐹2, 퐺2, 퐻2} be isotopisms from 푆1 to 푆 and 푆2 to 푆 respectively.

Let 푒1 and 푒2 denote the multiplicative identity elements of 푆1 and 푆2 respectively.

We will show that if 퐹1(푒1) = 퐹2(푒2) = 푦 and 퐺1(푒1) = 퐺2(푒2) = 푧, then 푆1 is 푛 isomorphic to 푆2. From there, since there are at most (푝 − 1) possible choices for 푦 and 푧, there can be at most (푝푛 − 1)2 nonisomorphic semiﬁelds isotopic to 푆.

Suppose 퐹1(푒1) = 퐹2(푒2) = 푦 and 퐺1(푒1) = 퐺2(푒2) = 푧, and let 퐿푦 and 푅푧 be the linear transformations for left and right multiplication by 푦 and 푧 respectively.

Then we have 퐿푦(푥) = 푦∗푥 = 퐹1(푒1)∗푥 = 퐹2(푒푥)∗푥 and 푅푧(푥) = 푥∗푧 = 푥∗퐺1(푒1) =

푥 ∗ 퐺2(푒2), and by Lemma 2.11 we get:

퐻1 = 푅푧 ∘ 퐹1 = 퐿푦 ∘ 퐺1

퐻2 = 푅푧 ∘ 퐹2 = 퐿푦 ∘ 퐺2

14 −1 −1 −1 −1 −1 Notice that (푅푧 ∘ 퐹2) = 퐹2 ∘ 푅푧 , and thus (푅푧 ∘ 퐹2) ∘ (푅푧 ∘ 퐹1) = 퐹2 ∘ 퐹1. By equivalence, we can then say

−1 −1 −1 퐻2 ∘ 퐻1 = 퐺2 ∘ 퐺1 = 퐹2 ∘ 퐹1 = 휙

Note that since 휙 is a composition of nonsingular linear transformations, it will be bijective and preserve addition. To show it is an isomorphism from 푆1 to 푆2 we only need to show that it preserves multiplication. By deﬁnition, 퐻2(휙(푎) ⋆2 휙(푏)) = −1 퐹2(휙(푎)) ∗ 퐺2(휙(푏)). By applying 퐻2 to both sides of this equation we get the desired results:

−1 휙(푎) ⋆2 휙(푏) = 퐻2 (퐹2(휙(푎)) ∗ 퐺2(휙(푏)))

−1 = 퐻2 (퐹1(푎) ∗ 퐺1(푏))

−1 = 퐻2 (퐻1(푎 ⋆1 푏))

= 휙(푎 ⋆1 푏)

In general, (푝푛 − 1)2 is an upper bound which is rarely attained. In the case where 푆 is a finite field, we will later prove that there are no semifields isotopic to 푆 which are not isomorphic to it. In regards to the examples previously given, System V is isotopic to 17 other nonisomorphic semifields, while System W is isotopic to 4 other nonisotopic semifields. The following result gives a specific means of constructing each of the (푝푛 − 1)2 semifields isotopic to a given semifield.

Theorem 2.13 (Knuth, [11]). Let 푆 be a semiﬁeld, 푦, 푧 ∈ 푆 be nonzero, and 퐿푦, 푅푧 be the linear transformations for left and right multiplication by 푦 and 푧 respectively.

15 −1 −1 ′ Let 퐹 = 푅푧 and 퐺 = 퐿푦 , and let 푆 be the additive group of 푆 with multiplication ⋆ deﬁned by

푎 ⋆ 푏 = 퐹 (푎) ∗ 퐺(푏)

Then 푆′ is a semiﬁeld isotopic to 푆 with multiplicative identity 푦 ∗ 푧. Furthermore, every semiﬁeld isotopic to 푆 can be constructed in this way (up to isomorphism).

Proof. We will start by showing that 푆′ satisfies the axioms of a semifield. The first axiom holds, since 푆′ was defined to be the additive group of 푆 with multiplication ⋆.

To show 푆′ has no zero divisors, suppose there exist 푎, 푏 ∈ 푆′ such that 푎⋆푏 = 0. Then

퐹 (푎) ∗ 퐺(푏) = 0, which implies 퐹 (푎) = 0 or 퐺(푏) = 0. Without loss of generality,

−1 ′ suppose 퐹 (푎) = 0. Then 푅푧 (푎) = 0, which is only possible if 푎 = 0. Thus 푆 has no zero divisors. To show that 푆′ is left distributive, consider

푎 ⋆ (푏 + 푐) = 퐹 (푎) ∗ (퐺(푏) + 퐺(푐)) = 퐹 (푎) ∗ 퐺(푏) + 퐹 (푎) ∗ 퐺(푐) = 푎 ⋆ 푏 + 푎 ⋆ 푐

A similar argument shows that 푆′ is right distributive. The ﬁnal axiom to test is whether 푆′ has a multiplicative identity, and we will now show that it will be 푦 ∗ 푧. First we show it is a right identity:

−1 −1 −1 푎 ⋆ (푦 ∗ 푧) = 퐹 (푎) ∗ 퐺(푦 ∗ 푧) = 푅푧 (푎) ∗ 퐿푦 (푦 ∗ 푧) = 푅푧 (푎) ∗ 푧 −1 = 푅푧(푅푧 (푎)) = 푎

Similarly, we show it is a left identity:

−1 −1 −1 (푦 ∗ 푧) ∗ 푎 = 퐹 (푦 ∗ 푧) ∗ 퐺(푎) = 푅푧 (푦 ∗ 푧) ∗ 퐿푦 (푎) = 푦 ∗ 퐿푦 (푎) −1 = 퐿푦(퐿푦 (푎)) = 푎

Thus, (푆′, +, ⋆) is a semiﬁeld isotopic to 푆 with isotopism {퐹 , 퐺, 퐼} from 푆′ to 푆, where 퐼 is the identity map.

To show that every semifield isotopic to 푆 can be defined in this way, we will use a result from the proof of the previous theorem. Let (푆1, +, ⋆1) be a semifield isotopic 16 to 푆, with isotopism {퐹1, 퐺1, 퐻1} from 푆1 to 푆. Let 푒1 be the multiplicative identity of 푆1, and 퐹1(푒1) = 푦 and 퐺1(푒1) = 푧. Now consider the semifield 푆2 constructed −1 −1 from 푆 via the isotopism {퐹2, 퐺2, 퐻2} from 푆2 to 푆, where 퐹2 = 푅푧 , 퐺2 = 퐿푦 and 퐻2 is the identity map. Then the multiplicative identity, 푒2, of 푆2 is equal to

푦 ∗ 푧, and we have 퐹2(푦 ∗ 푧) = 푦, 퐺2(푦 ∗ 푧) = 푧. Thus we have 퐹1(푒1) = 퐹2(푒2) and

퐺1(푒1) = 퐺2(푒2), and as shown in the proof of the previous theorem, 푆1 is isomorphic −1 −1 to 푆2 via the isomorphism 휙 = 퐹2 ∘ 퐹1 = 퐺2 ∘ 퐺1.

This theorem provides a very useful tool when looking at the isotopisms of

semiﬁelds. As mentioned at the start of this section, applying an isotopism to a

semiﬁeld could yield a system lacking a multiplicative identity! But, if the isotopism

−1 −1 consists of 퐹 = 푅푧 and 퐺 = 퐿푦 , then the resulting system will be a semiﬁeld.

2.6 The Geometric Signiﬁcance of Isotopisms

This section elaborates on the relationship between semiﬁelds and projective planes. In particular, we investigate the relationship between isomorphic projective planes and isotopic semiﬁelds.

Deﬁnition 2.14. Two projective planes, 휋 and 휋′ are said to be isomorphic if there

′ exists a bijection 휙 ∶ 휋 → 휋 which preserves incidence, i.e. if point (푥1, 푥2, 푥3) lies on line [푦1, 푦2, 푦3] in 휋, then 휙(푥1, 푥2, 푥3) lies on 휙[푦1, 푦2, 푦3]. An automorphism of a projective plane is called a collineation.

Lemma 2.15 (Knuth, [11]). Let 휋 and 휋′ be isomorphic semiﬁeld planes with isomorphism 휙 such that

휙(1, 0, 0) = (1, 0, 0) , 휙(0, 1, 0) = (0, 1, 0) , 휙(0, 0, 1) = (0, 0, 1)

Then the semiﬁelds which coordinatize 휋 and 휋′ are isotopic.

17 Proof. We use equation 2.10 to help deﬁne the points and lines of 휋. By hypothesis we have

휙[0, 0, 1] = 휙(0, 1, 0) ∶ 휙(0, 0, 1) = (0, 1, 0) ∶ (0, 0, 1) = [0, 0, 1]

Note that 휙(0, 1, 푎) lies on 휙[0, 0, 1] and thus must lie on [0, 0, 1]. Since [0, 0, 1] passes through (0, 1, 푎) and 휙(0, 1, 푎), there must exist a linear transformation 퐺 such that 휙(0, 1, 푎) = (0, 1, 퐺(푎)). This linear transformation will be the same 퐺 as in the isotopism {퐹 , 퐺, 퐻}, and in a similar way we will deﬁne 퐹 and 퐻. From the hypothesis we have

휙[0, 1, 0] = 휙(0, 0, 1) ∶ 휙(0, 1, 0) = (0, 0, 1) ∶ (0, 1, 0) = [0, 1, 0] and

휙[1, 0, 0] = 휙(0, 1, 0) ∶ 휙(1, 0, 0) = (0, 1, 0) ∶ (1, 0, 0) = [1, 0, 0]

Thus there must exist linear transformations 퐹 and 퐻 such that 휙(1, 푎, 0) = (1, 퐹 (푎), 0) and 휙(1, 0, 푏) = (1, 0, 퐻(푏)). We can then see that, in general

휙[0, 1, 푎] = 휙(0, 0, 1) ∶ 휙(1, −푎, 0) = (0, 0, 1) ∶ (1, −퐹 (푎), 푏) = [0, 1, 퐹 (푎)]

휙[1, 푎, 푏] = 휙(0, 1, 푎) ∶ 휙(1, 0, 푏) = (0, 1, 퐺(푎)) ∶ (1, 0, 퐻(푏)) = [1, 퐺(푎), 퐻(푏)]

휙(1, 푎, 푏) = 휙[0, 1, −푎] ∩ 휙[0, 1, 푏] = [0, 1, 퐹 (−푎)] ∩ [0, 1, 퐻(푏)] = (1, 퐹 (푎), 퐻(푏))

We can now show that the incidence relation is preserved. Recall that (푥1, 푥2, 푥3) lies on [푦1, 푦2, 푦3] if and only if 푦1푥3 = 푥2 ∗ 푦2 + 푥1푦3. We also know that 휙(푥1, 푥2, 푥3) lies on 휙[푦1, 푦2, 푦3], so we have the following relation, where the symbol “⇔” stands for “if and only if”:

푦1푥3 = 푥2 ∗ 푦2 + 푥1푦3 ⇔ 푦1퐻(푥3) = 퐹 (푥2) ∗ 퐺(푦2) + 푥1퐻(푦3)

18 Without loss of generality, we can let 푥1 = 푦1 = 1, as the cases where they equal zero will be similar. Then, by linearity, we get the following

푥3 = 푥2 ∗ 푦2 + 푦3 ⇔ 퐻(푥3) = 퐹 (푥2) ∗ 퐺(푦2) + 퐻(푦3)

⇔ 푥3 − 푦3 = 푥2 ∗ 푦2 ⇔ 퐻(푥3 − 푦3) = 퐹 (푥2) ∗ 퐺(푦2)

By substitution we have 퐻(푥2 ∗ 푦2) = 퐹 (푥2) ∗ 퐺(푦2), and thus {퐹 , 퐺, 퐻} is an isotopism.

Theorem 2.16 (Knuth, [11]). Let {퐹 , 퐺, 퐻} be an isotopism from a semiﬁeld 푆 to

a semiﬁeld 푆′, and let 휋 and 휋′ be the semiﬁeld planes coordinatized by 푆 and 푆′

respectively. Then 휋 and 휋′ are isomorphic.

Proof. We deﬁne 휙 ∶ 휋 → 휋′ using the relationships in the previous theorem, i.e.

휙[0, 0, 1] = [0, 0, 1] 휙(0, 0, 1) = (0, 0, 1)

휙[0, 1, 푎] = [0, 1, 퐹 (푎)] 휙(0, 1, 푎) = (0, 1, 퐺(푎))

휙[1, 푎, 푏] = [1, 퐺(푎), 퐻(푏)] 휙(1, 푎, 푏) = (1, 퐹 (푎), 퐻(푏))

From the previous lemma we know that 휙(1, 푥2, 푥3) ∈ 휙[1, 푦2, 푦3], but now we need to consider the cases where 푥1 = 0 or 푦1 = 0:

(0, 푥2, 푥3) ∈ [1, 푦2, 푦3] ⇔ 푥3 = 푥2 ∗ 푦2 ⇔ 푥2 = 1, 푦2 = 푥3

⇔ 퐺(푦2) = 퐺(푥2) ⇔ 휙(0, 푥2, 푥3) ∈ 휙[1, 푦2, 푦3]

(1, 푥2, 푥3) ∈ [0, 푦2, 푦3] ⇔ −푦3 = 푥2 ∗ 푦2 ⇔ 푦2 = 1, 푥2 = −푦3

⇔ 퐹 (푥2) = −퐹 (푦3) ⇔ 휙(1, 푥2, 푥3) ∈ 휙[0, 푦2, 푦3]

(0, 푥2, 푥3) ∈ [0, 푦2, 푦3] ⇔ 푥2 = 0 or 푦2 = 0

⇔ 휙(0, 푥2, 푥3) ∈ 휙[0, 푦2, 푦3]

19 This theorem is suﬃcient to show that isotopic semiﬁelds coordinatize isomor-

phic projective planes, which is suﬃcient to show the geometric motivations for study-

ing semiﬁelds. The converse is also true, as shown by Albert in [3], but the proof

requires developing more geometric tools, and is beyond the scope of this work.

2.7 Autotopisms and Automorphisms

An autotopism of a semiﬁeld 푆 is an isotopism, {퐹 , 퐺, 퐻}, between 푆 and itself, i.e. 퐻(푎 ∗ 푏) = 퐹 (푎) ∗ 퐺(푏). The set of all autotopisms of 푆 will clearly form a group under composition, which we will denote At(푆). Furthermore, if 푆 and 푆′ are isotopic, then |At(푆)| = |At(푆′)|, since each autotopism of 푆′ is determined by applying the isotopism from 푆 to 푆′ to the autotopisms of 푆.

In the case where 퐹 = 퐺 = 퐻, the autotopism is clearly an automorphism, and if we let Aut(푆) denote the automorphism group of 푆, then Aut(푆) ⊂ At(푆).

Lemma 2.17 (Knuth, [11]). Let 휋 be a semiﬁeld plane coordinatized by a semiﬁeld 푆.

Then all semiﬁelds isotopic to 푆 coordinatize 휋, and the collineations of 휋 which ﬁx

(0, 0, 1), (0, 1, 0), and (1, 0, 0) form a group isomorphic to the autotopism group of 푆.

Proof. This follows directly from Lemma 2.15 and Theorem 2.16.

This leads to an interesting result which has both algebraic and geometric sig-

niﬁcance.

Theorem 2.18 (Knuth, [11]). Let 푆 be a semiﬁeld of order 푝푛. Then

|At(푆)| (2.13) (푝푛 − 1)2 = ∑ |Aut(푆′)| 푆′

where 푆′ ranges over all nonisomorphic semiﬁelds isotopic to 푆.

Proof. Let 푦 and 푧 range over the nonzero elements of 푆 and let 푆′ be one of the (푝푛 −

1)2 semiﬁelds isotopic to 푆 determined by a choice of 푦 and 푧 as described in Theorem 20 2.13. We will show that there are |At(푆)|/|Aut(푆′)| such isotopes isomorphic to 푆′, which will prove the theorem.

′ −1 −1 ′ Let 푆 be isomorphic and isotopic to 푆, with isotopism {푅푧 , 퐿푦 , 퐼} from 푆 to 푆, where 퐼 is the identity map, and isomorphism 휙 ∶ 푆′ → 푆. Then 휙 is an automorphism of 푆, and {휙 ∘ 푅푧, 휙 ∘ 퐿푦, 휙} is an autotopism of 푆. Every autotopism of 푆 will have this form and will be determined (up to isomorphism) by 푦 and 푧.

Suppose 푟 choices of 푦 and 푧 yield isomorphic semiﬁelds. Then |At(푆)| = 푟|Aut(푆)|.

For any semiﬁeld 푆′ which is isotopic to 푆 but not isomorphic to 푆, we will similarly have |At(푆′)| = 푟|Aut(푆′)|. Since |At(푆′)| = |At(푆)|, we have |At(푆)| =

푟|Aut(푆′)|. In each case, 푟 is the number of isotopes of 푆 isomorphic to a particular isotope 푆′, and thus the sum of all such 푟 must equal (푝푛 −1)2. This gives the desired result.

21 Chapter 3

Cubical Arrays

In chapter 2, we showed that any semiﬁeld 푆 is an 푛-dimensional vector space

over a ﬁnite ﬁeld 픽푝. Let {푥1, 푥2, ..., 푥푛} be a basis of 푆. Then there is a 푛 × 푛 × 푛

set of scalars 퐴 = {퐴푖푗푘}, deﬁned by

푛

(3.1) 푥푖 ∗ 푥푗 = ∑ 퐴푖푗푘푥푘 푘=1 This set, 퐴, is an example of a cubical array, and in this chapter we will provide a formal definition for cubical arrays and show how vectors and matrices can act upon them. We will then provide a number of results regarding the relationship between cubical arrays and finite semifields. All of the results and proofs in this chapter are taken from [11] and are included here for the sake of completeness. Some of the notation has been changed to reflect modern conventions and to avoid ambiguity. We have taken a more focused approach to these results than was taken in [11], so many of the proofs in this section have been modified from what appears in [11].

3.1 Cubical Array Basics

Definition 3.1. A cubical array, 퐴 =, of dimension 푛 over a finite field, 픽푞, is an

푛 × 푛 × 푛 ordered set of elements of 픽푞. The (푖, 푗, 퐾)-th entry of 퐴 is denoted 퐴푖푗푘 where 1 ≤ 푖, 푗, 푘 ≤ 푛.

We will use a notation similar to 퐴푖푗푘 for vectors and matrices. Given a vector

푣, we will let 푣푖 denote the 푖-th entry of 푣, and given a matrix 푀, we will let 푀 푖푗 denote the (푖, 푗)-th entry of 푀. When working with cubical arrays it is often useful

22 to construct matrices from their entries. Given a cubical array 퐴, we deﬁned the

following matrices by ﬁxing an index of 퐴:

(3.2) 퐴푖∗∗ = 푀 ⇔ 퐴푖푗푘 = 푀 푗푘

(3.3) 퐴∗푗∗ = 푀 ⇔ 퐴푖푗푘 = 푀 푖푘

(3.4) 퐴∗∗푘 = 푀 ⇔ 퐴푖푗푘 = 푀 푖푗 Knuth deﬁnes a similar notation in [11] which uses superscripts in place of subscripts.

푖∗∗ It is important to note that our notation diﬀers slightly though, with 퐴푖∗∗ = 퐴 ∗∗푘 ∗푗∗ 푇 ∗푗∗ 푇 ∗푗∗ and 퐴∗∗푘 = 퐴 , but 퐴∗푗∗ = (퐴 ) , where (퐴 ) ) is the transpose of 퐴 . We will sometimes wish to construct vectors from the rows and columns of matrices, so

we use a similar notation for that as well.

(3.5) 푀푖∗ = 푣 ⇔ 푀 푖푗 = 푣푗

(3.6) 푀∗푗 = 푣 ⇔ 푀 푖푗 = 푣푖 Note then that by ﬁxing two of the indices of 퐴, we get a vector. Thus 퐴 can be

thought of as a three-dimensional array, a set of 푛 square 푛 × 푛 matrices, or an 푛 × 푛

matrix whose entries are 푛-dimensional vectors. This notation also eliminates the need for distinguishing between row and column vectors. For the purposes of this

work, all vectors will be assumed to be row vectors, and thus we will have matrices

act on vectors by right multiplication, i.e. 푣푀. Finally, we will also deﬁne the product of a vector and a cubical as follows:

푛

(3.7) 푣퐴 = ∑ 푣푖퐴푖∗∗ 푖=1

Deﬁnition 3.2. A cubical array, 퐴, is said to be nonsingular over a ﬁeld, 픽푞, if, for all 푛 nonzero 푣 ∈ 픽푞 , the matrix 푛

푀 = ∑ 푣퐴푖∗∗ 푖=1 is nonsingular. 23 Deﬁnition 3.3. A cubical array, 퐴, is said to be in standard form if 퐴1∗∗ = 퐴∗1∗ = 퐼,

where 퐼 is the 푛 × 푛 identity matrix over 픽푞. These two deﬁnitions will prove to be of immense value in the following theorem.

푛 Consider a semiﬁeld, 푆, of order 푝 , let 풳 = {푥1, ..., 푥푛} be a basis of 푆 over 픽푝, and let 푎, 푏 ∈ 푆. Assuming we know some method for constructing 푆, such as those given

in equations 2.6, 2.7, and 2.8, we then have two ways of viewing 푎 and 푏: the forms

described by the construction and the coordinate vectors with respect to 풳. Since

it will often be necessary to consider both views, we will use 푎 and 푏 to denote the

vector forms of 푎 and 푏 respectively. If 퐴 is the cubical array deﬁned by 3.1, we say

퐴 is the cubical array corresponding to 푆 generated by 풳. Let 푐 = 푎 ∗ 푏. Then, by the distributive property of semiﬁelds, we have

푛 푛 푛 푛 푛 푛 푛

푐 = 푎 ∗ 푏 = (∑ 푎푖푥푖) ∗ (∑ 푏푗푥푗) = ∑ ∑ 푎푖푏푗푥푖 ∗ 푥푗 = ∑ ∑ ∑ 푎푖푏푗퐴푖푗푘푥푘 푖=1 푗=1 푖=1 푗=1 푖=1 푗=1 푘=1

Thus the multiplication in 푆 is completely determined by 퐴. It is also worth noting

푛 that addition in 푆 corresponds to standard vector addition in 픽푝 , where 0 corresponds to the zero vector. We can now prove the following result, which gives necessary and

sufficient conditions for a cubical array to define a semifield.

푛 Theorem 3.4 (Knuth, [11]). Let 푆 be a semiﬁeld of order 푝 , and let 풳 = {푥1, ..., 푥푛}

be a basis for 푆 over 픽푝. Then the cubical array 퐴 corresponding to 푆 generated

by 풳 will be nonsingular, and will be in standard form if and only if 푥1 is the multiplicative identity. Conversely, if 퐴 is a nonsingular cubical array in standard form, then equation 3.1 deﬁnes an 푛-dimensional semiﬁeld over 픽푝 using a formal

basis {푥1, ..., 푥푛}, where 푥1 will be the multiplicative identity.

Proof. First, let 퐴 be a cubical array corresponding to 푆 generated by 풳, and let

푛 푣, 푤 ∈ 픽푝 be nonzero. Let 푀 = 푣퐴. Then 푀 is nonsingular if and only if 푤푀 is

24 nonzero. There exist nonzero elements 푣, 푤 ∈ 푆 corresponding to these vectors. Since

푆 has no zero divisors, 푢 = 푣 ∗ 푤 is nonzero. Then 푤푀 is nonzero by the following: 푛 푛 푛 푛 푛

푢 = ∑ ∑ 푣푖푤푗퐴푖푗∗ = ∑ 푤푗 (∑ 푣푖퐴푖푗∗) = ∑ 푤푗푀 푗∗ = 푤푀 푖=1 푗=1 푗=1 푖=1 푗=1 Therefore 퐴 is nonsingular. Notice in the previous argument that 푣 ∗ 푤 = 푤푀. Let

푒 be the multiplicative identity of 푆, and let 퐹 = ∑푛 푒 퐴 . Then 푒 ∗ 푤 = 푤퐹 , 푖=1 푖 푖∗∗ but since 푒 ∗ 푤 = 푤, 푒 ∗ 푤 = 푤, and 퐹 must be the identity matrix. In a similar way, consider the following:

푛 푛 푛 푛 푛

푤 ∗ 푒 = ∑ ∑ 푤푖푒푗퐴푖푗∗ = ∑ 푤푖 (∑ 푒푗퐴푖푗∗) = ∑ 푤푖퐺푖∗ = 푤퐺 푖=1 푗=1 푖=1 푗=1 푖=1

In this case, 푤 ∗ 푒 = 푤퐺 = 푤 so 퐺 must be the identity matrix as well. If 푒 = 푥1,

then 푒1 = 1, and 푒푖 = 0 for 푖 > 1. Plugging this into the deﬁnitions of 퐹 and 퐺 we have 퐹 = ∑푛 푒 퐴 = 퐴 = 퐼 푖=1 푖 푖∗∗ 1∗∗ 퐺 = ∑푛 푒 퐴 = 퐴 = 퐼 푗=1 푗 푖푗∗ ∗1∗ Thus 퐴 is in standard form. This also shows that the converse is true as well, since the nonsingularity of a cubical array will ensure that no zero divisors exist, and being

in standard form ensures the existence of a multiplicative identity.

푛 Corollary 3.5 (Knuth, [11]). Let 푆 be a semiﬁeld of order 푝 , 풳 = {푥1, ..., 푥푛} be a

basis of 푆 over 픽푝, and 퐴 be the cubical array corresponding to 푆 generated by 풳.

Let 푎, 푏 ∈ 푆. Then the linear transformations 퐿푎 and 푅푏 have matrix representations with respect to 풳, 퐿 and 푅, defined by 푛 푛 푇 (3.8) 퐿 = ∑ 푎푖퐴푖∗∗ = 푎퐴 푅 = ∑ 푏푗퐴∗푗∗ = 푏퐴 푖=1 푗=1 Notice that we have introduced a new piece of notation in this corollary: the transpose of 퐴. To clarify its meaning, and the fact that 푏퐴푇 will still yield a nonsingular matrix, we have the following. 25 Definition 3.6. Let 퐴 be a cubical array. The cubical array 퐵 defined by 퐵푖푗푘 = 퐴푗푖푘 is called the transpose of 퐴 and is denoted 퐴푇 .

Lemma 3.7 (Knuth, [11]). Let 퐴 be a cubical array. Then 퐴푇 is nonsingular and in standard form if and only if 퐴 is nonsingular and in standard form.

Proof. If 퐴 is nonsingular and in standard form, it deﬁnes a semiﬁeld, 푆, as previously

푇 푛 discussed. Let 퐵 = 퐴 and let 푣 ∈ 픽푝 be nonzero. Then there exists a nonzero

푣 ∈ 푆 corresponding to 푣, which has a linear transformation 푅푣 with matrix form 푅. Since 푅 is a linear transformation, 푅 is nonsingular, and thus ∑푛 푣 퐴 is 푣 푗=1 푗 ∗푗∗ nonsingular. By deﬁnition, 퐴 = 퐵 , and we have ∑푛 푣 퐵 is nonsingular, ∗푗∗ 푗∗∗ 푗=1 푗 푗∗∗

thus 퐵 is nonsingular. Further, 퐼 = 퐴1∗∗ = 퐵∗1∗ and 퐼 = 퐴∗1∗ = 퐵1∗∗, so 퐵 is in standard form.

The transpose of a cubical array has further significance. Let 휋 be a semifield plane coordinatized by 푆. It is well-known that the dual of 휋 is coordinatized by a semifield, 푆′, which is anti-isomorphic to 푆. This yields the following definition.

Definition 3.8. Let (푆, +, ∗) be a semifield. A semifield (푆′, +, ⋆) is said to be the dual of 푆 if there exists a nonsingular linear transformation 휙 ∶ 푆 ↦ 푆′ such that

휙(푎 ∗ 푏) = 휙(푏) ⋆ 휙(푎).

Lemma 3.9 (Knuth, [11]). Let 퐴 be a cubical array corresponding to a semiﬁeld

′ 푆 generated by a basis 풳 = {푥1, ..., 푥푛}, and let 푆 be the dual of 푆, with anti- isomorphism 휙 ∶ 푆 ↦ 푆′. Then 퐴푇 is the cubical array corresponding to 푆′ generated by 휙(풳) = {휙(푥1), ..., 휙(푥푛)}.

Proof. Applying 휙 to equation 3.1 gives us

휙(푥푖 ∗ 푥푗) = 휙 (∑ 퐴푖푗푘푥푘) = ∑ 퐴푖푗푘휙(푥푘) = 휙(푥푗) ⋆ 휙(푥푖) 푘=1 푘=1

26 Let 퐵 = 퐴푇 . Then we have

휙(푥푗) ⋆ 휙(푥푖) = ∑ 퐴푖푗푘휙(푥푘) ⇔ 휙(푥푖) ⋆ 휙(푥푗) = ∑ 퐵푖푗푘휙(푥푘) 푘=1 푘=1

As noted at the beginning of this chapter, these results are all slightly specialized

cases of results that Knuth proved in [11]. In honor of his work and the integral role

that Theorem 3.4 will play in the following work, we give the following deﬁnition.

Definition 3.10. If 퐴 is a nonsingular cubical array in standard form, then 퐴 will be called a Knuth cube. If 푆 is a semifield of order 푝푛 we will let 퐾(푆) denote the set of all 푛 × 푛 × 푛 Knuth cubes over 픽푝 which define 푆. Note that, since 푛 and 푝 are finite, then there are only a finite number of

푛 × 푛 × 푛 cubical arrays over 픽푝, and thus 퐾(푆) must be ﬁnite as well. We will further elaborate on this in future chapters, and eventually provide the size of 퐾(푆).

3.2 Examples

Recall from section 2.3 that the three examples of 16-element semiﬁelds all had

2 elements of the form 푎 + 휆푏 where 푎, 푏 ∈ 픽4 with the convention 픽4 = {0, 1, 휔, 휔 = 1 + 휔}. The multiplication in the three semiﬁelds was deﬁned by

픽16 (푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏푑휔) + 휆(푏푐 + 푎푑 + 푏푑) System W (푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏2푑휔) + 휆(푏푐 + 푎2푑)

System V (푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏2푑) + 휆(푏푐 + 푎2푑 + 푏2푑2)

All of these semiﬁelds are 4-dimensional vector spaces over 픽2, and a suitable basis for Knuth cubes for all three of these semiﬁelds is 풳 = {1, 휔, 휆, 휆휔}. Since the cubical arrays are three-dimensional, we will express them as matrices whose entries

27 are vectors, i.e. 푀 푖푗 = 퐴푖푗∗, and we will write the vectors as strings of digits. Then the cubical array generated by 풳 for each semiﬁeld is:

⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ 0100 1100 0001 0011 ⎟ ⎜ ⎟ 픽16 퐹 = ⎜ ⎟ ⎜ 0010 0001 0110 1101 ⎟ ⎜ ⎟ ⎝ 0001 0011 1101 1011 ⎠

⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ System W 푊 = ⎜ ⎟ ⎜ 0010 0001 0100 1100 ⎟ ⎜ ⎟ ⎝ 0001 0011 1000 0100 ⎠

⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ System V 푉 = ⎜ ⎟ ⎜ 0010 0001 1010 0111 ⎟ ⎜ ⎟ ⎝ 0001 0011 1111 1001 ⎠

This is how Knuth presented cubical arrays in [11], and this representation has a number of beneﬁts. First, consider the matrices contained in 푊 . In order to construct

푊2∗∗, we look at the vectors which make up the second row of 푊 . These vectors will form the rows of 푊2∗∗, and we have

⎛ 0 1 0 0 ⎞ ⎜ ⎟ ⎜ 1 1 0 0 ⎟ ⎜ ⎟ 푊2∗∗ = ⎜ ⎟ ⎜ 0 0 1 1 ⎟ ⎜ ⎟ ⎝ 0 0 1 0 ⎠

28 Finding 푊∗2∗ is also straightforward, as the rows of 푊∗2∗ will be the vectors which make up the second column of 푊 :

⎛ 0 1 0 0 ⎞ ⎜ ⎟ ⎜ 1 1 0 0 ⎟ ⎜ ⎟ 푊∗2∗ = ⎜ ⎟ ⎜ 0 0 0 1 ⎟ ⎜ ⎟ ⎝ 0 0 1 1 ⎠

To ﬁnd 푊∗∗푘, we build a matrix whose (푖, 푗)-th entry is the 푘-th entry of 푊푖푗∗. For instance, 푊∗∗2 is obtained from the second entry in each vector in 푊 :

⎛ 0 1 0 0 ⎞ ⎜ ⎟ ⎜ 1 1 0 0 ⎟ ⎜ ⎟ 푊∗∗2 = ⎜ ⎟ ⎜ 0 0 1 1 ⎟ ⎜ ⎟ ⎝ 0 0 0 1 ⎠ This representation also makes the transpose of a cubical array more intuitive, as the matrix form of 푊 푇 will be the transpose of the matrix form of 푊 . Notice that

푇 퐹 = 퐹 , which corresponds to the fact that 픽16 is commutative.

3.3 The Triple Product

So far we have only deﬁned the product of a vector and a cubical array. In [11],

Knuth devotes a great deal of work to deﬁning a general class of products between

arrays of arbitrary dimension, and the vector and cubical array product is a special

case of this. For our purposes there is only one other case of Knuth’s products which

we will need.

29 Deﬁnition 3.11. Let 퐴 be an 푛 × 푛 × 푛 cubical array over 픽푞, and let 퐹 , 퐺, and 퐻 be 푛 × 푛 matrices over 픽푞. Then the triple product of [퐹 , 퐺, 퐻] and 퐴 is the cubical array 퐵 deﬁned by 푛 푛 푛

퐵푖푗푘 = ∑ ∑ ∑ 퐹 푖푟퐺푗푠퐻푘푡퐴푟푠푡 푟=1 푠=1 푡=1 and is denoted 퐵 = [퐹 , 퐺, 퐻] × 퐴

Lemma 3.12 (Knuth, [11]). Let 퐴 be a cubical array of dimension 푛 over 픽푞, and let

퐵, 퐶, 퐷, 퐹 , 퐺, and 퐻 be 푛 × 푛 matrices over 픽푞. Then the following equation is true:

(3.9) [퐵, 퐶, 퐷] × ([퐹 , 퐺, 퐻] × 퐴) = [퐵퐹 , 퐶퐺, 퐷퐻] × 퐴

Proof. The proof is straightforward, based on the deﬁnition of the triple product and matrix multiplication. First, recall that (퐵퐹 ) = ∑푛 퐵 퐹 . Let 퐵 = 푥푦 푧=1 푥푧 푧푦 [퐹 , 퐺, 퐻] × 퐴 and 퐶 = [퐵, 퐶, 퐷] × 퐵. Then

푛 푛 푛

퐶푖푗푘 = ∑ ∑ ∑ 퐵푖푟퐶푗푠퐷푘푡퐵푟푠푡 푟=1 푠=1 푡=1

푛 푛 푛 푛 푛 푛

= ∑ ∑ ∑ 퐵푖푟퐶푗푠퐷푘푡 (∑ ∑ ∑ 퐹 푟푥퐺푠푦퐻푡푧퐴푥푦푧) 푟=1 푠=1 푡=1 푥=1 푦=1 푧=1

푛 푛 푛 푛 푛 푛

= ∑ ∑ ∑ ∑ ∑ ∑ 퐵푖푟퐹 푟푥퐶푗푠퐺푠푦퐷푘푡퐻푡푧퐴푥푦푧 푟=1 푠=1 푡=1 푥=1 푦=1 푧=1

푛 푛 푛

= ∑ ∑ ∑(퐵퐹 )푖푥(퐶퐺)푗푦(퐷퐻)푘푧퐴푥푦푧 푥=1 푦=1 푧=1

= [퐵퐹 , 퐶퐺, 퐷퐻] × 퐴

30 Lemma 3.13 (Knuth, [11]). Let 퐴 be a cubical array of dimension 푛 over 픽푞, let 퐹 ,

퐺, and 퐻 be 푛 × 푛 matrices over 픽푞, and let 퐵 = [퐹 , 퐺, 퐻] × 퐴. Then the following formulas are true

푛 푇 (3.10) 퐵푖∗∗ = 퐺 (∑ 퐹푖푟퐴푟∗∗) 퐻 푟=1 푛 푇 (3.11) 퐵∗푗∗ = 퐹 (∑ 퐺푗푠퐴∗푠∗) 퐻 푠=1 푛 푇 (3.12) 퐵∗∗푘 = 퐹 (∑ 퐻푘푡퐴∗∗푡) 퐺 푡=1 Proof. These formulas are also a direct consequence of the deﬁnition of the triple product and matrix multiplication. For example, to derive the ﬁrst equation we start with

푛 푛 푛 푛 푛 푛

퐵푖푗푘 = ∑ ∑ ∑ 퐹 푖푟퐺푗푠퐻푘푡퐴푟푠푡 ⇔ 퐵푖∗∗ = ∑ 퐹 푖푟 (∑ ∑ 퐺∗푠퐴푟푠푡퐻∗푡) 푟=1 푠=1 푡=1 푟=1 푠=1 푡=1

For any particular 푟 we will let 푀 = 퐴푟∗∗, and note that 퐺푠∗퐻∗푡 = (퐺퐻)푠푡. Now 푇 consider (퐺푀)퐻 . By the deﬁnition of matrix multiplication we have

푛

(퐺푀)푥푦 = ∑ 퐺푥푠푀 푠푦 푠=1 Then 푛 푛 푛 푇 푇 ((퐺푀)퐻 )푥푦 = ∑(퐺푀)푥푡퐻푡푦 = ∑ ∑ 퐺푥푠푀 푠푡퐻푦푡 푡=1 푡=1 푠=1 And thus 푛 푛 푇 ((퐺푀)퐻 ) = ∑ ∑ 퐺∗푠푀 푠푡퐻∗푡 푡=1 푠=1

Recall that 푀 = 퐴푟∗∗, so 푀 푠푡 = 퐴푟푠푡, which gives the desired result, by substitution:

푛 푛 푛 푛 푛 푡 푇 퐵푖∗∗ = ∑ 퐹 푖푟 (∑ ∑ 퐺∗푠퐴푟푠푡퐻∗푡) = ∑ 퐹 푖푟(퐺퐴푟∗∗퐻 ) = 퐺 (∑ 퐹푖푟퐴푟∗∗) 퐻 푟=1 푠=1 푡=1 푟=1 푟=1 The other formulas can be derived in a similar way. 31 Throughout this work we will use nonsingular 푛 × 푛 matrices over 픽푝, and the following deﬁnition provides a useful shorthand for such matrices.

Deﬁnition 3.14. The group 퐺퐿(푛, 푞) is the set of all 푛 × 푛 invertible matrices with

entries in 픽푞 together with standard matrix multiplication. Theorem 3.15 (Knuth, [11]). Let 퐴 be a nonsingular cubical array of dimension 푛

over 픽푞, and let 퐹 , 퐺, 퐻 ∈ 퐺퐿(푛, 푞). Then 퐵 = [퐹 , 퐺, 퐻] × 퐴 is also nonsingular.

푛 Proof. Let 푣 ∈ 픽푞 be nonzero. We need to show that 푣퐵 is nonsingular. By equation 3.10, we have

푛 푛 푛 푛 푛 푇 푇 푣퐵 = ∑ 푣푖퐵푖∗∗ = ∑ 푣푖 (퐺 (∑ 퐹푖푟퐴푟∗∗) 퐻 ) = 퐺 (∑ ∑ 푣푖퐹 푖푟퐴푟∗∗) 퐻 푖=1 푖=1 푟=1 푖=1 푟=1

Since both 퐺 and 퐻 are nonsingular, the only way 푣퐵 is singular is if ∑푛 ∑푛 푣 퐹 퐴 푖=1 푟=1 푖 푖푟 푟∗∗ is singular. Note that 푣퐹 = ∑푛 푣 퐹 , so if 푤 = 푣퐹 , then 푤 = ∑푛 푣 퐹 . Since 푖=1 푖 푖∗ 푟 푖=1 푖 푖푟 퐹 is nonsingular and 푣 is nonzero, 푤 is nonzero. Since 퐴 is nonsingular, 푤퐴 is

nonsingular, and thus 퐵 is nonsingular since

푛 푛 푛

∑ ∑ 푣푖퐹 푖푟퐴푟∗∗ = ∑ 푤푟퐴푟∗∗ 푖=1 푟=1 푟=1

3.4 Isotopism

In this section we present a few crucial theorems which show how the cubical

arrays of isotopic semiﬁelds are related.

Theorem 3.16 (Knuth, [11]). Let 푆 and 푆′ be isotopic semiﬁelds of order 푝푛 with isotopism {퐹 , 퐺, 퐻} from 푆′ to 푆. Let 퐴 be a cubical array corresponding to 푆

generated by a basis 풳 = {푥1, ..., 푥푛}. Let 퐹 , 퐺, 퐻 ∈ 퐺퐿(푛, 푝) be the matrix forms −푇 of 퐹 , 퐺, and 퐻 with respect to 풳. Then 퐵 = [퐹 , 퐺, 퐻 ] × 퐴 is a cubical array

corresponding to 퐵 generated by 풳. 32 Proof. Note that 퐹 , 퐺, and 퐻 are deﬁned by the following:

푛 푛 푛

퐹 (푥푖) = ∑ 퐹 푖푟푥푟 퐺(푥푗) = ∑ 퐺푠푗푥푠 퐻(푥푘) = ∑ 퐻푘푡푥푡 푟=1 푠=1 푡=1 Let ⋆ denote the multiplication in 푆′, and let 퐵 be the cubical array corresponding to 푆′ generated by 풳, i.e. 푛

푥푖 ⋆ 푥푗 = ∑ 퐵푖푗푘푥푘 푘=1 By the deﬁnition of isotopism, we also have

퐻(푥푖 ⋆ 푥푗) = 퐹 (푥푖) ∗ 퐺(푥푗)

Then we have

−1 푥푖 ⋆ 푥푗 = 퐻 (퐹 (푥푖) ∗ 퐺(푥푗))

= 퐻−1 ((∑푛 퐹 푥 ) ∗ (∑푛 퐺 푥 )) 푟=1 푖푟 푟 푠=1 푗푠 푠

= 퐻−1 (∑푛 ∑푛 퐹 퐺 푥 ∗ 푥 ) 푟=1 푠=1 푖푟 푗푠 푟 푠

= 퐻−1 (∑푛 ∑푛 ∑푛 퐹 퐺 퐴 푥 ) 푟=1 푠=1 푡=1 푖푟 푗푠 푟푠푡 푡

−1 = ∑푛 ∑푛 ∑푛 ∑푛 퐹 퐺 퐻 퐴 푥 푟=1 푠=1 푡=1 푘=1 푖푟 푗푠 푡푘 푟푠푡 푘

−푇 Then 퐵 = [퐹 , 퐺, 퐻 ] × 퐴.

푛 Lemma 3.17 (Knuth, [11]). Let 푆 be a semiﬁeld of order 푝 and let 풳 = {푥1, ..., 푥푛}

and 풴 = {푦1, ..., 푦푛} be bases of 푆 over 픽푝. Let 퐴 and 퐵 be the cubical arrays generated by 풳 and 풴 respectively, and let 퐶 be the change of basis matrix from 풳 to 풴. Then −푇 퐵 = [퐶, 퐶, 퐶 ] × 퐴 33 Proof. Note that 퐶 satisﬁes the following equations

푛 푛 −1 푦푖 = ∑ 퐶푖푗푥푗 푥푖 = ∑ 퐶푖푗 푦푗 푗=1 푗=1

From this we have

푦 ∗ 푦 = (∑푛 퐶 푥 ) ∗ (∑푛 퐶 푥 ) 푖 푗 푟=1 푖푟 푟 푠=1 푗푠 푠

= ∑푛 ∑푛 퐶 퐶 푥 ∗ 푥 푟=1 푠=1 푖푟 푗푠 푟 푠

= ∑푛 ∑푛 ∑푛 퐶 퐶 퐴 푥 푟=1 푠=1 푡=1 푖푟 푗푠 푟푠푡 푡

−1 = ∑푛 ∑푛 ∑푛 ∑푛 퐶 퐶 퐴 퐶 푦 푟=1 푠=1 푡=1 푘=1 푖푟 푗푠 푟푠푡 푡푘 푘

= ∑푛 퐵 푦 푘=1 푖푗푘 푘

−푇 where, by deﬁnition of the triple product, 퐵 = [퐶, 퐶, 퐶 ] × 퐴.

We now have the means to construct a Knuth cube for any isotope of a semiﬁeld.

Theorem 3.18 (Knuth, [11]). Let 푆 and 푆′ be isotopic semiﬁelds, and let 퐴 ∈ 퐾(푆).

Then there exists 퐵 ∈ 퐾(푆′) such that

−1 −1 −푇 (3.13) 퐵 = [퐶푅 , 퐶퐿 , 퐶 ] × 퐴

where 푅, 퐿, and 퐶 are the matrices corresponding to 푅푧, 퐿푦, and 퐿푦∗푧 respectively for some nonzero 푦, 푧 ∈ 푆.

Proof. This is a direct consequence of a number of previous theorems. By Theorem

−1 −1 2.13 there exist nonzero 푦, 푧 ∈ 푆 such that {푅푧 , 퐿푦 , 퐼} is an isotopism between ′ 푆 and 푆 . Let 풳 = {푥1, ..., 푥푛} be the basis which generates 퐴, and let 푅, 퐿, and 퐼 be the matrices corresponding to 푅푧, 퐿푦 and 퐼 with respect to 풳. Then 34 −1 −1 −푇 퐷 = [푅 , 퐿 , 퐼 ] × 퐴 will be a cubical array corresponding to 푆′ by Theorem

3.16. Note that 퐷 may not be in standard form, so we need to apply a change of

basis from 풳 to an ordered basis whose ﬁrst element is the multiplicative identity

of 푆′. By Theorem 2.13, the multiplicative identity of 푆′ is 푦 ∗ 푧, so the matrix

corresponding to 퐿푦∗푧 would be a suitable choice. Let 퐶 be the matrix corresponding to 퐿푦∗푧 with respect to 풳. Since 퐴 is in standard form, 푥1 is the multiplicative identity of 푆. Let 풴 = {푦1, ..., 푦푛} be deﬁned by

푛

푦푖 = ∑ 퐶푖푗푥푗 푗=1

Since 퐿푦∗푧(푥1) = 푦 ∗ 푧, 퐶1∗ is the vector corresponding to 푦 ∗ 푧 with respect to 풳, −푇 which in turn is equal to 푦1. Now let 퐵 = [퐶, 퐶, 퐶 ] × 퐷 will be a nonsingular cubical array in standard form corresponding to 푆′. Then 퐵 ∈ 퐾(푆′) and

−푇 −1 −1 −푇 −1 −1 −푇 퐵 = [퐶, 퐶, 퐶 ] × ([푅 , 퐿 , 퐼 ] × 퐴) = [퐶푅 , 퐶퐿 , 퐶 ] × 퐴

35 Chapter 4

The Relationship Between 푆 and 픽푝[푥]

There does not yet exist a strong theory regarding polynomials and semifields, and the use of polynomials in the study of semifields has been limited to the minimal and characteristic polynomials of the matrices for left and right multiplication by semifield elements. Albert, [3], was probably the first to utilize the minimal polynomials of these matrices, which he used to prove the existence of a cyclic basis for semifields of order 16 and 32. Oehmke, [14], used both the minimal and characteristic polynomials to prove commutativity of a specific class of semifields. More recently, two independent proofs of the following result were presented by Rúa, Cambarro, and

Ranilla in [18], and Gow and Sheekey in [6].

Theorem. Let 푆 be a semiﬁeld, 푎 ∈ 푆, and 퐿 be the matrix associated with left multi-

plication by 푎. Then 푎 is a left primitive element of 푆 if and only if the characteristic

polynomial of 퐿 is primitive. The work in this chapter is an attempt to build a foundation for the study of the relationship between elements of a finite semifield and the polynomial ring over an appropriate finite field. The overall structure and motivation for this work comes from Chapter 3 of Finite fields and their applications [12], which discusses the properties of polynomials over finite fields. And, as a direct consequence of these investigations, we provide a new proof for the previously mentioned theorem.

36 4.1 Exponents, Primitive Elements, and Cyclic Bases

As we mentioned in chapter 2, standard exponential notation is not well deﬁned

for semiﬁeld elements due to the nonassociativity of multiplication. A common way

of addressing this issue is to deﬁne two new exponential notations which refer to

repeated multiplication on the left or right. For example, if 푎 ∈ 푆, we use the following inductive deﬁnition.

(4.1) 푎(1 = 푎푎(푖+1 = 푎 ∗ 푎(푖

(4.2) 푎1) = 푎푎푖+1) = 푎푖) ∗ 푎

Due to the fact that semiﬁelds are both left and right distributive, it is common practice to present results using only the left or right multiplication. We will do the same, and focus entirely on left multiplication. The primary reason for choosing left multiplication instead of right is simply because, if 퐴 ∈ 퐾(푆) and 푎 is the vector form

of 푎 with respect to 퐴, then the matrix for left multiplication by 푎 is 푎퐴. From this point on, all of the deﬁnitions and results presented will be based on left multiplication

by an element. We will continue to use the term left in order to emphasize this choice

and the fact that similar results would hold if right multiplication were used.

Deﬁnition 4.1. Let 푆 be a semiﬁeld of order 푝푛. An element 푎 ∈ 푆 is said to be left

primitive if, for all nonzero 푏 ∈ 푆, there exists 1 ≤ 푖 ≤ 푝푛 − 1 such that 푎(푖 = 푏. It is well known that finite fields contain primitive elements, but finite semifields are a different matter altogether. First, note that we have defined left primitive elements, and similarly we could define right primitive elements. As will be illustrated in chapter 8, the left and right primitive elements of a semifield can be different.

Wene conjectured in [22] that every semifield contained a right primitive element, but a counterexample was provided by Rúa, [17], of a semifield which was neither left nor right primitive. Thus we cannot assume a semifield contains left primitive 37 elements, but the existence of left primitive elements in semifields is common enough

to warrant study.

푛 Deﬁnition 4.2. Let 푆 be a semiﬁeld of order 푝 . Let 푎 ∈ 푆 be nonzero, and let ℒ푎 be the set of left powers of 푎 from 0 to 푛 − 1, i.e.

(1 (푛−1 (4.3) ℒ푎 = {1, 푎 , ..., 푎 }

If ℒ푎 is a basis of 푆 over 픽푝, then ℒ푎 is called the left cyclic basis generated by 푎. Currently, there is no proof that such a basis exists for every semiﬁeld, but there has not yet been a semiﬁeld found which does not possess such a basis. Oehmke provides a possible partial proof that such a basis exists in [14], by showing that, for

푛 a semiﬁeld, 푆, of order 푝 the indeterminate vector 푥 = (푥1, ..., 푥푛) generates a right cyclic basis of the algebra 푆 ⊗ 픽푝[푥1, ..., 푥푛]. Thus, the matrix 푀 whose 푖-th row is 푛 the (푖 + 1)-st right power of 푥 is nonsingular, and if we substitute 푣 ∈ 픽푝 for 푥 in 푀, then we will get a matrix whose rows correspond to the vector form of the ﬁrst

푛 − 1 right powers of some element of 푆. Note that, if there exists a 푣 for which 푀 remains nonsingular, then that would prove the existence of a right cyclic basis for

푆. By taking the determinant of 푀, we get a polynomial 푓(푥1, ..., 푥푛), which may 푛 not be homogeneous, and we would need to show that there exists 푣 ∈ 픽푝 such that 푓(푣) ≠ 0.

4.2 Left Order and Minimal Polynomial

We begin by deﬁning properties of semiﬁeld elements analogous to those of

elements of ﬁnite ﬁelds. It is useful to consider the matrices for left multiplication

by semiﬁeld elements, so we will introduce a new notation which works well with our

existing vector notation. Consider a semiﬁeld 푆 of order 푝푛 and 퐴 ∈ 퐾(푆) deﬁned

by a basis 풜 = {1, 푥2, ..., 푥푛}. For any 푎 ∈ 푆, 푎 is the vector corresponding to 푎 38 with respect to 풜, and we will let 퐿푎 denote the matrix for left multiplication by 푎 with respect to 풜. Note that 1 is the ﬁrst vector of the standard basis, (1, 0, 0, ..., 0),

which is commonly denoted 푒1. Then we also have 푎퐴 = 퐿푎 and

푎 = 푎 ∗ 1 = 푎 ∗ 1 = 1퐿푎 = 푒1퐿푎

We can now begin investigating the left multiplicative properties of semiﬁeld elements.

Lemma 4.3. For any 푎 ∈ 푆, there exists an integer 푘 ≤ 푝푛 − 1 such that 푎(푘 = 1.

(푖 Proof. Consider the set (푎) = {푎 ; 푖 ∈ ℤ+} of all positive left powers of 푎. As noted (푖 (푖 푛 lemma 2.2, for any 푎 , there exists 푏푖 such that 푏푖푎 = 푎. There are, at most, 푝 − 1 (푖 (푖 distinct 푏푖. If this is the case, then for some 푖 we have 푎푎 = 푎, in which case 푎 = 1. 푛 푛 If there are fewer than 푝 − 1 distinct 푏푖, then there exists 푗 < 푖 < 푝 − 1 such that (푖 (푗 (푖 (푗 (푖 (푗 (푖−푗 푏푖푎 = 푏푖푎 , which gives 푏푖(푎 −푎 ) = 0, which forces 푎 −푎 = 0 and 푎 = 1.

Deﬁnition 4.4. For 푎 ∈ 푆, the smallest integer 푘 such that 푎(푘 = 1 is called the left order of 푎 and denoted ord푙(푎).

푛 푘 Lemma 4.5. Let 푎 ∈ 푆. Then there is a positive integer 푘 ≤ 푝 − 1 such that 퐿푎 = 퐼, where 퐼 is the identity matrix.

Proof. Let 푓(푥) be the characteristic polynomial of 퐿푎, i.e. 푓(푥) = |퐿푎 − 푥퐼|. Since

퐿푎 is nonsingular, we have deg(푓) = 푛. Note that, by the Cayley-Hamilton theorem,

푓(퐿푎) = 0 and 푓(0) ≠ 0. By lemma 3.1 in [12], there exists a positive integer 푛 푘 푘 푘 푘 ≤ 푝 − 1 such that 푓(푥)|(푥 − 1). Thus 퐿푎 − 퐼 = 0, and 퐿푎 = 퐼.

푘 Deﬁnition 4.6. The smallest integer 푘 such that 퐿푎 = 퐼 is called the order of 퐿푎, and

denoted ord(퐿푎). 푙 Lemma 4.7. Let 푎 ∈ 푆. Then ord (푎)|ord(퐿푎).

Proof. Let ord푙(푎) = 푘. Then we have the following

(푘 푘 푎 = 1 = 푒1퐿푎 = 푒1 39 푘 (푘+1 푘+1 So the ﬁrst row of 퐿푎 is 푒1. Now consider 푎 = 푒1퐿푎 = 푎. In other words, the

푘+1 푘+푖 (푖 first row of 퐿푎 is 푎. In general, the first row of 퐿푎 will be 푎 . Thus, the first row 푗 푙 of 퐿푎 is 푒1 if and only if 푗 is a multiple of 푘. And so ord (푎)|ord(퐿푎).

Notice that, in the case of finite fields, these results are trivial. If 푎 ∈ 픽푝푛 , then 푙 푘 ord (푎) = ord(푎), and if ord(푎) = 푘, then (퐿푎) = 퐿푎푘 = 퐼. Thus ord(푎) = ord(퐿푎). These lemmas provide a strong foundation regarding the exponential properties of the elements of semifields. Given a polynomial 푓 ∈ 픽푝[푥] and 푎 ∈ 푆, we define

푓(푎) in the natural way. More explicitly, for 푐푖 ∈ 픽푝, we have

푗 푗 푖 (푖 (4.4) 푓(푥) = 푐0 + ∑ 푐푖푥 ⇒ 푓(푎) = 푐0(1) + ∑ 푐푖푎 푖=1 푖=1 The nonassociative multiplication in 푆 causes some trouble when evaluating polynomials, since polynomial multiplication assumes the indeterminate is self-associative.

Some polynomials can be factored into a product of lesser degree polynomials, and

this could cause 푓(푎) to not be well deﬁned. For example, consider the polynomial

4 2 푓(푥) = 푥 +푥 +1 in 픽2[푥], and the element 휆 in system W. Recall that multiplication in system W is deﬁned by

(푎 + 휆푏) ∗ (푐 + 휆푑) = (푎푐 + 푏2푑휔) + 휆(푏푐 + 푎2푑)

So 휆2 = 휔, 휆(3 = 휆휔, and 휆(4 = 1 + 휔. Then 푓(휆) = (1 + 휔) + 휔 + 1 = 0. But

푓(푥) = (푥2 + 푥 + 1)2, and

(휆2 + 휆 + 1)2 = (1 + 휔 + 휆)2 = 휔 + 휔 + 휆(1 + 휔 + 휔) = 휆

Thus, it is worth emphasizing that when a polynomial is evaluated at a semifield element, it is evaluated in its fully expanded form as written in equation 4.4. We can now define when a semifield element is a root of a polynomial, and begin looking at the relationship between polynomials and semifield elements. 40 Definition 4.8. Let 푓 ∈ 픽푝[푥], 푎 ∈ 푆, and 푓(푎) be defined by equation 4.4. If 푓(푎) = 0, then 푎 is called a left root of 푓.

Lemma 4.9. Let 푎 ∈ 푆 and 푓, 푔, ℎ ∈ 픽푞[푥] such that 푓 + 푔 = ℎ. Then 푓(푎) + 푔(푎) = ℎ(푎).

Proof. Let 푑 be the degree of ℎ. Then, there exist 푏푖, 푐푖 ∈ 픽푝, such that

푑 푑 푛 푖 푖 푖 푓(푥) = ∑ 푏푖푥 푔(푥) = ∑ 푐푖푥 ℎ(푥) = ∑(푏푖 + 푐푖)푥 푖=0 푖=0 푖=0 Since 푆 is distributive, we have

푑 푑 푑 (푖 (푖 (푖 푓(푎) + 푔(푎) = (∑ 푏푖푎 ) + (∑ 푐푖푎 ) = ∑(푏푖 + 푐푖)푎 = ℎ(푎) 푖=0 푖=0 푖=0

Lemma 4.10. Let 푎 ∈ 푆 and 푓, 푔, ℎ ∈ 픽푝[푥] such that 푓푔 = ℎ and 푎 is a left root of 푓 or 푔. Then 푎 is a left root of ℎ.

Proof. This proof is not as simple as at may at ﬁrst appear. As already mentioned we must look at the ﬁnal form of ℎ and show that 푎 is a left root of that form. First, let 푓 and 푔 have the following forms:

푗 푘 푖 푖 푓(푥) = ∑ 푏푖푥 푔(푥) = ∑ 푐푖푥 푖=0 푖=0

Without loss of generality, suppose 푎 is a left root of 푓. Note that ℎ(푥) = ∑푘 푐 푥푖푓(푥), 푖=0 푖 and, by the previous lemma, 푎 is a left root of ℎ if and only if 푎 is a left root of the

푖 expanded form of 푐푖푥 푓(푥) for all 푖. Now consider

푖 푖 푖+1 푖+푗 푓푖(푥) = 푐푖푥 푓(푥) = 푐푖푏0푥 + 푐푖푏1푥 + ⋯ + 푐푖푏푗푥

41 By un-distributing multiples of 푎 we obtain the desired result:

(푖 (푖+1 (푖+푗 푓푖(푎) = 푐푖푏0푎 + 푐푖푏1푎 + ⋯ + 푐푖푏푗푎 (푖−1 (푖 (푖+푗−1 = 푐푖푎(푏0푎 + 푏1푎 + ⋯ + 푏푗푎 ) ⋮

(푗 = 푐푖푎(푎(⋯ 푎(푏0 + 푏1푎 + ⋯ + 푏푗푎 ) ⋯)

= 푐푖푎(푎(⋯ 푎(0) ⋯) = 0

Thus 푓푖(푎) = 0 for all 푖 and ℎ(푎) = 0.

Lemma 4.11. Let 푎 ∈ 푆. Then there exists a unique monic polynomial of minimal

degree 푓 ∈ 픽푝[푥] such that 푎 is a left root of 푓.

Proof. Since there exists 푘 ≤ 푝푛 − 1 with 푎(푘 = 1, 푎 is a left root of 푥푘 − 1. Thus 푎 is

a left root of a monic polynomial in 픽푝[푥]. Let 푛 be the lowest degree of polynomial

in 픽푝[푥] for which 푎 is a left root. Suppose 푓 and 푔 are monic polynomials of degree 푛 with 푎 as a left root, with

푛−1 푛−1 푛 푖 푛 푖 푓(푥) = 푥 + ∑ 푏푖푥 푔(푥) = 푥 + ∑ 푐푖푥 푖=0 푖=0 Since 푓(푎) = 푔(푎) = 0, we have the following:

푎(푛 + ∑푛−1 푏 푎(푖 = 푎(푛 + ∑푛−1 푐 푎(푖 푖=0 푖 푖=0 푖

⇒ ∑푛−1 푏 푎(푖 = ∑푛−1 푐 푎(푖 푖=0 푖 푖=0 푖

⇒ ∑푛−1(푏 − 푐 )푎(푖 = 0 푖=0 푖 푖

Let ℎ(푥) = ∑푛−1(푏 − 푐 )푥푖. Note that, since ℎ has degree less than 푛, 푎 cannot be 푖=0 푖 푖 a left root of ℎ unless ℎ(푥) = 0. Thus 푏푖 = 푐푖 for all 푖. 42 Deﬁnition 4.12. Let 푎 ∈ 푆. The unique monic polynomial of minimal degree in 픽푝[푥] for which 푎 is a left root will be called the left minimal polynomial of 푎.

As mentioned in chapter 2, if 푆 is a semiﬁeld of order 푝푛, then 푆 contains a

subﬁeld isomorphic to 픽푝 consisting of scalar multiples of the multiplicative identity. It is sometimes useful to identify the elements of 푆 which are not in this subﬁeld, and we will refer to such elements as being nonscalar. The following lemma and corollary

present some useful properties of the left minimal polynomial of a nonscalar semiﬁeld

element.

Lemma 4.13. Let 푎 ∈ 푆 be nonscalar, and let 푓 ∈ 픽푝[푥] be a monic polynomial for

which 푎 is a left root. If there exist polynomials 푔, ℎ ∈ 픽푝[푥] such that 푓 = 푔ℎ and ℎ(푥) = 푥 − 푐, then 푎 is a left root of 푔.

Proof. Note that 푔 can be written as 푔(푥) = ∑푛−1 푏 푥푖, with 푏 = 1. Then, 푓 has 푖=0 푖 푛−1 the form

푛−1 푛 푛−1 푖 푖+1 푖 푓(푥) = ∑ 푏푖푥 (푥 − 푐) = (∑ 푏푖푥 ) + (∑(−푐)푏푖푥 ) 푖=0 푖=0 푖=0 Since 푎 is a left root of 푓, we then have

푛 푛−1 푛−1 (푖+1 (푖 (푖−1 푓(푎) = (∑ 푏푖푎 ) + (∑(−푐)푏푖푎 ) = (푎 − 푐) (∑ 푏푖푎 ) = 0 푖=0 푖=0 푖=0 Note that 푎 − 푐 ≠ 0 so 푔(푎) = 0, and 푎 is a left root of 푔.

Corollary 4.14. Let 푎 ∈ 푆 be nonscalar, and let 푓 ∈ 픽푝[푥] be the left minimal polynomial of 푎. Then 푓 has no linear factors, 푓(푐) ≠ 0 for all 푐 ∈ 픽푝, and deg(푓) < ord(푓). Consequently, 푓 is either irreducible or the product of irreducible polynomials of at least second degree.

43 4.3 Determining the Left Minimal Polynomial

The left minimal polynomial of a semiﬁeld element can be determined using

a method similar to the one used in [12] to ﬁnd the minimal polynomial of a ﬁnite

ﬁeld element. Let 푆 be a semiﬁeld of order 푝푛, 푎 ∈ 푆, and let 푓 be the left minimal

polynomial of 푎. Clearly deg 푓 ≤ 푛, otherwise 푆 would contain a set of more than 푛 linearly independent elements. Then 푓 has the following form, for 푐푖 ∈ 픽푝, assuming some trivial 푐푖, 푛 푓(푥) = 푐푛푥 + ⋯ + 푐1푥 + 푐0

Then 푎 is a left root of 푓 if and only if

푛 (푖 ∑ 푐푖푎 = 0 푖=0

Let 푀 be the (푛 + 1) × 푛 matrix whose 푖-th row is 푎(푖−1. Let 푟 be the rank of 푀.

Then, if 푐 = (푐0, 푐1, ..., 푐푛), 푓(푎) = 푐푀. By the Rank-Nullity theorem, if the rank of 푀 is 푟 and dimension of the set of solutions is 푠, then 푠 = 푛 + 1 − 푟. Since 1 ≤ 푟 ≤ 푛,

1 ≤ 푠 ≤ 푛. Thus, if 푠 coordinates of 푐 are prescribed, the other coordinates are uniquely determined. If 푠 = 1, set 푐푛 = 1. If 푠 > 1, set 푐푛 = 푐푛−1 = ⋯ = 푐푛−푠+1 = 0,

and 푐푛−푠 = 1. This remaining values of 푐푖 will deﬁne 푓, and the degree of 푓 will be 푟. For example, consider the element 휆 in system W. We will look at the vector

coordinates of the powers of 휆 with respect to the basis {1, 휔, 휆, 휆휔}. As we’ve

previously mentioned, 휆2 = 휔, 휆(3 = 휆휔, and 휆(4 = 1+휔. Then, 푀 has the following form: ⎛ 1 0 0 0 ⎞ ⎜ ⎟ ⎜ 0 0 1 0 ⎟ ⎜ ⎟ 푀 = ⎜ 0 1 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 1 ⎟ ⎜ ⎟ ⎝ 1 1 0 0 ⎠ 44 Then we have 푟 = 4 and 푠 = 1. Let 푐 = (푐0, 푐1, 푐2, 푐3, 1), and set 푐푀 = 0. This gives

(푐0 + 1, 푐2 + 1, 푐1, 푐3) = (0, 0, 0, 0), which gives 푐0 = 푐2 = 1 and 푐1 = 푐3 = 0. Thus, the left minimal polynomial for 휆 is 푓(푥) = 푥4 + 푥2 + 1.

4.4 Consequences of the Left Minimal Polynomial

The existence of the left minimal polynomial has many applications. As previ-

ously mentioned, if 푎 is a left root of a polynomial 푓, then 푎 may not be a left root of

any factor of 푓. The following lemma shows how the left minimal polynomial helps address this issue.

Lemma 4.15. Let 푎 ∈ 푆 and let 푓, 푔 ∈ 픽푝[푥] with 푓 the left minimal polynomial of 푎. Then 푓|푔 if and only if 푎 is a left root of 푔.

Proof. If 푓|푔, then, by lemma 4.10, 푔(푎) = 0. Suppose 푎 is a left root of 푔. If

deg(푓) = deg(푔), then 푔 is a scalar multiple of 푓 and thus 푎 is clearly a left root

of 푔. Suppose deg(푓) < deg(푔). By the division algorithm, there exist ℎ, 푟 ∈ 픽푝[푥] such that 푔 = ℎ푓 + 푟, with deg(푟) < deg(푓). Let 푞 = ℎ푓. By lemmas 4.9 and 4.10,

푞(푎) = 0, 푔(푎) = 푞(푎) + 푟(푎), and we have 0 = 푟(푎). But deg(푟) < deg(푓) and 푓 is

the minimal polynomial of 푎, so 푟(푥) = 0. Thus 푓|푔.

Since 퐿푎 ∈ 퐺퐿(푛, 푝), it is well known that there exists a minimal polynomial

for 퐿푎 which must divide its characteristic polynomial. And since 푓(푎) = 푒1푓(퐿푎),

푎 will be a left root of both the minimal and characteristic polynomials of 퐿푎. The following lemma provides a suﬃcient condition for the minimal polynomials of 푎 and

퐿푎 to be equal. Lemma 4.16. Let 푎 ∈ 푆. Let 푓 be the minimal polynomial of 푎 and let 푔 be the

푙 minimal polynomial of 퐿푎. Then 푓|푔, and 푓 = 푔 if and only if ord (푎) = ord(퐿푎).

45 Proof. As mentioned, 푔(푎) = 0, so, by lemma 4.15, 푓|푔. Thus there exists ℎ ∈ 픽푝[푥] such that 푔 = ℎ푓. Since 푔(퐿푎) = 0, either 푓(퐿푎) = 0 or ℎ(퐿푎) = 0. Note that the degrees of 푓 and ℎ must be less than or equal to the degree of 푔, but that 푔 is the

minimal polynomial of 퐿푎. Thus 푓 or ℎ must be trivial. Since 푓 is not trivial, ℎ must 푙 be, and we have 푓 = 푔. Conversely, if 푓 = 푔, then clearly ord (푎) = ord(퐿푎).

Note that an element 푎 ∈ 푆 will deﬁne a cyclic basis if and only if its left

minimal polynomial has degree 푛. This yields the following.

Corollary 4.17. Let 푎 ∈ 푆, and let 퐿푎 be the matrix for left multiplication by 푎 with

respect to some basis. If the characteristic polynomial of 퐿푎 is irreducible, then 푎 deﬁnes a cyclic basis.

Recall that, for 푓 ∈ 픽푝[푥], the order of 푓, denoted ord(푓), is the smallest integer 푘 such that 푓|(푥푘 − 1). For 푎 ∈ 푆, if 푓 is the left minimal polynomial of 푎,

and 푘 = ord푙(푎), then by deﬁnition 푎(푘 = 1, 푎 is a left root of 푥푘 − 1, and we have

푓|(푥푘 − 1). This yields the following important result.

Theorem 4.18. Let 푎 ∈ 푆 and let 푓 be the left minimal polynomial of 푎. Then

ord푙(푎) = ord(푓). This rather straightforward result allows us to study the algebraic properties of

a semiﬁeld element by studying its left minimal polynomial. The following theorem

summarizes a number of results that follow directly from the previous theorem and

results from [12] regarding the order of a polynomial.

Theorem 4.19. Let 푎 ∈ 푆, let 푓 be the left minimal polynomial of 푎, and let ord푙(푎) =

푘. Then the following are true:

1. If 푓 is irreducible and deg(푓) = 푛, then 푘|(푝푛 − 1).

2. Let 푐 be a positive integer. Then 푓|(푥푐 − 1) if and only if 푘|푐.

46 푏 3. If 푓 = 푔 where 푏 is a positive integer and 푔 ∈ 픽푝[푥] is irreducible with 푔(0) ≠ 0 and ord(푔) = 푐. Then 푘 = 푐푝푡, where 푡 is the smallest integer such that 푝푡 ≥ 푏.

4. If 푔1, ..., 푔푛 ∈ 픽푝[푥] are pairwise relatively prime, and 푓 = 푔1 ⋯ 푔푛, then 푘 =

lcm(ord(푔1), ..., ord(푔푛)).

푐1 푐푛 5. If 푓 = 푏푓1 ⋯ 푓푛 , where 푏 ∈ 픽푝, 푐1, ..., 푐푛 are positive integers, and 푓1, ..., 푓푛 ∈ 푡 픽푝[푥] are distinct monic irreducible polynomials, then 푘 = 푑푝 where 푑 =

lcm(ord(푓1), ..., ord(푓푛)), 푡 is the smallest positive integer greater than or equal

to max(푐1, ..., 푐푛). 푛 6. 푓 is a primitive polynomial of degree 푛 over 픽푝 if and only if 푘 = 푝 − 1. This theorem provides some useful tools for studying semifield elements. For example, if the left minimal polynomial of 푎 is irreducible, then 푎(푝푛 = 푎. In the case of a finite field this is always true, but in a semifield, the order of an element may not divide 푝푛 − 1. We conclude by proving the result from the beginning of this chapter, which follows almost directly from the previous results.

Corollary 4.20. Let 푎 ∈ 푆 and let 푓 be the left minimal polynomial of 푎. Then 푎 is left primitive if and only if 푓 is a left primitive polynomial of degree 푛 over 픽푝. Theorem 4.21. An element 푎 ∈ 푆 is left primitive if and only if the characteristic polynomial of 퐿푎 is primitive.

Proof. Let 푓 be the left minimal polynomial of 푎 and 푔 the characteristic polynomial of 퐿푎. First, suppose 푎 is left primitive. Then 푓 is primitive, which implies that 푓 is irreducible and deg(푓) = 푛. Also, 푓 must divide the minimal polynomial of 퐿푎,

which must divide 푔, so 푓|푔. Since 퐿푎 is an 푛 × 푛 matrix, deg(푔) = 푛 as well, so

푓 = 푔. Thus the characteristic polynomial of 퐿푎 is primitive. Now, suppose 푔 is

47 primitive. Then 푔 is irreducible and 푓 must divide 푔. Thus 푓 = 푔 and 푓 is a primitive polynomial of degree 푛 over 픽푝. Thus 푎 is left primitive.

48 Chapter 5

Determining Aut(푆)

Presently, all of the results regarding the automorphism groups of semifields have been found for very specific cases. Wene ([23], [24]) has determined the automorphism groups of semifields following particular constructions. Al-ali, [1], has

4 determined the automorphism group of semiﬁelds of order 푞 which admit ℤ2 × ℤ2 as an automorphism group which acts freely on the semiﬁeld, where 푞 > 3 is a prime power. The main results in this chapter provide a tool which can be used to determine

the automorphism group of any semiﬁeld using a brute-force approach.

5.1 Investigating 퐾(푆)

Theorem 3.4 states that any Knuth cube of dimension 푛 over 픽푝 can deﬁne a

semiﬁeld using a formal basis, 풳 = {푥1, ..., 푥푚}. Note that the coordinate vector

for 푥푖 with respect to 풳 will be 푒푖, the 푖-th standard basis vector. For this reason, rather than using a Knuth cube to deﬁne a semiﬁeld over a formal basis, it is more

푛 convenient to simply define a semifield product on 픽푝 by using the Knuth cube to define the products of the elements of the standard basis.

푛 Definition 5.1. Let 푆 be a semifield of order 푝 , 퐴 ∈ 퐾(푆), and let {푒1, ..., 푒푛} denote 푛 푛 the standard basis of 픽푝 . Then ∗퐴 denotes the semifield product on 픽푝 defined by 푛

(5.1) 푒푖 ∗퐴 푒푗 = ∑ 퐴푖푗푘푒푘 푘=1 푛 In this case, 푆 = (픽푝 , +, ∗퐴), where + denotes standard vector addition, and

푒1 is the multiplicative identity. Note that, viewing 푆 in this way, any other cube 푛 in 퐾(푆) must be deﬁned by an ordered basis of 픽푝 , 풴 = {푦1 = 푒1, 푦2, ..., 푦푛}. The 49 change of basis matrix, 퐶, from the standard basis to 풴 will have the form 퐶1∗ = 푦푖. This leads us to the following results.

Lemma 5.2. Let 퐶(푛, 푝) denote the set of all 푋 ∈ 퐺퐿(푛, 푝) whose ﬁrst row is 푒1. Then 퐶(푛, 푝) is a subgroup of 퐺퐿(푛, 푝).

Proof. Clearly the identity matrix, 퐼, is in 퐶(푛, 푝). Let 푋, 푌 ∈ 퐶(푛, 푝). Since 푌 ∈ −1 −1 −1 퐺퐿(푛, 푝), 푌 exists, and 푌 푌 = 퐼. By assumption 푌 1∗ = 푒1, so 푒1푌 = 퐼1∗ = 푒1, −1 −1 −1 and thus 푌 1∗ = 푒1, and 푌 ∈ 퐶(푛, 푝). If 푍 = 푋푌 , then 푍 ∈ 퐶(푛, 푝) by the following −1 −1 푍1∗ = 푋1∗푌 = 푒1푌 = 푒1

Theorem 5.3. Let 푆 be a semiﬁeld of order 푝푛, 퐴 ∈ 퐾(푆), and 퐶 ∈ 퐶(푛, 푝). Then −푇 [퐶, 퐶, 퐶 ] × 퐴 deﬁnes a group action of 퐶(푛, 푝) on 퐾(푆).

−푇 Proof. By Lemma 3.17 we know that 퐵 = [퐶, 퐶, 퐶 ] × 퐴 deﬁnes 푆. We now show

that 퐵 is in standard form. From equation 3.10, , we have

푛 −1 −1 퐵1∗∗ = 퐶 (∑ 퐶1푖퐴푖∗∗) 퐶 = 퐶퐼퐶 푖=1 Similarly, by equation 3.11,

푛 −1 −1 퐵∗1∗ = 퐶 (∑ 퐶1푗퐴∗푗∗) 퐶 = 퐶퐼퐶 = 퐼 푗=1

Thus 퐵 is in standard form, and we have 퐵 ∈ 퐾(푆). Finally, for 퐶, 퐷 ∈ 퐶(푛, 푝), we need −푇 −푇 [퐶, 퐶, 퐶 ] × ([퐷, 퐷, 퐷 ] × 퐴) = [퐶퐷, 퐶퐷, (퐶퐷)−푇 ] × 퐴

−푇 −푇 −1 −1 Note that 퐶 퐷 = (퐷 퐶 )푇 = (퐶퐷)−푇 . Then this is true by Theorem 3.12.

50 푛 Consider the results of this theorem in terms of building semiﬁelds from 픽푝 .

Let 풜 and ℬ be distinct bases of 푆 over 픽푝, and 퐴, 퐵 ∈ 퐾(푆) the Knuth cubes deﬁned by 풜 and ℬ respectively. Let ∗퐴 and ∗퐵 denote the multiplication deﬁned by 퐴 and 퐵 acting on the standard basis. Then, there is a linear transformation 퐶 −푇 corresponding to the matrix 퐶 ∈ 퐶(푛, 푝) satisfying 퐵 = [퐶, 퐶, 퐶 ] × 퐴, such that

∀푎, 푏 ∈ 푆,

퐶(푎 ∗퐴 푏) = 퐶(푎) ∗퐵 퐶(푏)

푛 푛 푛 And we have 퐶 ∶ (픽푝 , +, ∗퐴) ↦ (픽푝 , +, ∗퐵) is an isomorphism. Note that (픽푝 , +, ∗퐴) = 푛 (픽푝 , +, ∗퐵) if and only if 퐴 = 퐵. Thus we have proven the following result. 푛 푛 Theorem 5.4. Let 푆 be a semiﬁeld of order 푝 , 퐴 ∈ 퐾(푆), and 푆 = (픽푝 , +, ∗퐴). Then −푇 퐶 ∈ 퐶(푛, 푝) is an automorphism of 푆 if and only if 퐴 = [퐶, 퐶, 퐶 ] × 퐴.

In practice, suppose we have a semiﬁeld 푆 of order 푝푛. We ﬁrst generate a

푛 Knuth cube 퐴 for 푆, and give 푆 the form (픽푝 , +, ∗퐴). Then, Aut(푆) is the subgroup of 퐶(푛, 푝) which ﬁxes 퐴, i.e. the stabilizer of 퐴. This fact allows us to determine the size of 퐾(푆).

Lemma 5.5. |퐶(푛, 푝)| = ∏푛−1(푝푛 − 푝푖). 푖=1

Proof. This is a straightforward result due to the fact that, if 퐶 ∈ 퐶(푛, 푝), then

퐶1∗ = 푒1 and the remaining rows must be linearly independent.

퐴 ∈ 퐾(푆). Then, for each 휙 ∈ Aut(푆), 휙(풜) is a distinct basis which also deﬁnes

퐴.

51 5.2 Utilizing Properties of Automorphisms

Clearly 퐶(푛, 푝) can be quite large, and, even with computer assistance, a brute force approach to determining Aut(푆) from 퐶(푛, 푝) could take an impractically long time. The work in this section aims to address this problem by signiﬁcantly reducing

the number of matrices which need to be tested. Recall that, for 푧 ∈ 푆, the notation

푧(푖 refers to repeated left multiplication.

Lemma 5.7. Let 푆 be a semiﬁeld of order 푝푛, 푧 ∈ 푆, and 휙 ∈ Aut(푆). Then 휙(푧(푖) =

(푖 휙(푧) . Also, if ℒ푧 is a basis of 푆, then so is ℒ휙(푥).

Proof. By deﬁnition of the left powers of 푧 and of automorphisms, we have

휙(푧(푖) = 휙(푧 ∗ 푧(푖−1) = 휙(푧) ∗ 휙(푧(푖−1) = 휙(푧) ∗ (휙(푧) ∗ ⋯ ∗ (휙(푧) ∗ 휙(푧)) ⋯) = 휙(푧)(푖

Corollary 5.8. Let 푧 ∈ 푆, 휙 ∈ Aut(푆). Then the following are true.

1. If 푓 is the left minimal polynomial of 푧, then 푓 is the left minimal polynomial

of 휙(푧).

2. If 푧 is left primitive, then 휙(푧) is left primitive.

3. If ℒ푧 is a basis of 푆, then ℒ휙(푧) is also a basis of 푆. This corollary provides a means by which we can reduce the number of matrices

which need to be tested. Consider a semiﬁeld, 푆, of order 푝푛 and 퐴 ∈ 퐾(푆). First,

determine the characteristic polynomial, 푓푧, of 푧퐴 for all nonzero 푧 ∈ 푆. Make

a list, 풫, of all 푧 for which 푓푧 is primitive, and a list, ℐ, of all 푧 for which 푓푧 is irreducible. Note that, as mentioned in chapter 4, there is no guarantee that 풫 or ℐ

will actually contain any elements, but the empirical evidence suggests that ℐ should

not be empty. Suppose 풫 is not empty. Then pick a particular 푧 ∈ 풫 with left

minimal polynomial 푓 and make a new list 풫′ of all 푦 ∈ 풫 which also have 푓 as their

52 (푖−1 left minimal polynomial. Let 퐶 be deﬁned by 퐶푖∗ = 푧 , and construct the cubical −푇 array 퐵 = [퐶, 퐶, 퐶 ] × 퐴. At this point 퐵 is the cubical array for 푆 generated by

′ ℒ푧. For all 푦 ∈ 풫 , 푦퐶 will be the vector for 푦 with respect to ℒ푧. Thus, for each

′ (푖−1 푦 ∈ 풫 , let 퐷 be the matrix deﬁned by 퐷푖∗ = (푦 퐶)푖∗. Then, determine which −푇 퐷 satisfy 퐵 = [퐷, 퐷, 퐷 ] × 퐵. These matrices will be the automorphisms of 푆 with respect to ℒ푧. If 풫 is empty, then a similar approach using ℐ will yield similar results. If both 풫 and ℐ are empty, then some other means of reducing the number of matrices to test must be found.

We provide an application of this method in section 5.4. For now, note that it should reduce the number of matrices to check from |퐶(푛, 푝)| to 푝푛 − 푝. The only case where this may not work is if the semiﬁeld in question does not have a cyclic basis.

5.3 Automorphisms of System V

System V is an interesting example because it has 6 automorphisms, which is more than any other 16-element semifield, including 픽16. Recall that the elements 2 in system V have the form 푎 + 휆푏, where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔}. Addition is defined in the standard way, and multiplication is defined by

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏2푑2)

Then 풜 = {1, 휔, 휆, 휆휔} is a basis for system V over 픽2 which generates the following Knuth cube: ⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0111 ⎟ ⎜ ⎟ ⎝ 0001 0011 1111 1001 ⎠ 53 Since |퐶(4, 2)| = 1, 344, it is a simple matter to ﬁnd all 퐶 in 퐶(4, 2) such that −푇 [퐶, 퐶, 퐶 ] × 퐴 = 퐴. By computation, the following matrices were found:

⎛ 1 0 0 0 ⎞ ⎛ 1 0 0 0 ⎞ ⎛ 1 0 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ 1. ⎜ ⎟ 2. ⎜ ⎟ 3. ⎜ ⎟ ⎜ 0 0 1 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 0 1 ⎠ ⎝ 0 0 1 1 ⎠ ⎝ 0 0 1 0 ⎠

⎛ 1 0 0 0 ⎞ ⎛ 1 0 0 0 ⎞ ⎛ 1 0 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 0 0 ⎟ ⎜ 1 1 0 0 ⎟ ⎜ 1 1 0 0 ⎟ 4. ⎜ ⎟ 5. ⎜ ⎟ 6. ⎜ ⎟ ⎜ 0 0 1 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 1 1 ⎠ ⎝ 0 0 1 0 ⎠ ⎝ 0 0 0 1 ⎠

4 If we view system V as (픽2, +, ∗퐴), then each of these matrices is actually an automorphism. Additionally, we can use these matrices to find more algebraically significant definitions of the automorphisms. We will let 휙푖 denote the automorphism corresponding to the 푖-th matrix listed above. Recall that the rows of these matrices are the vectors of the images of the basis elements under the automorphisms. For example, the second matrix gives the following information about 휙2:

1 0 0 0 휙 (1) = 1 ⎜⎛ ⎟⎞ 2 ⎜ ⎟ ⎜ 0 1 0 0 ⎟ 휙2(휔) = 휔 ⎜ ⎟ ⇒ ⎜ ⎟ ⎜ 0 0 0 1 ⎟ 휙2(휆) = 휆휔 ⎜ ⎟ ⎝ 0 0 1 1 ⎠ 휙2(휆휔) = 휆 + 휆휔

2 Since 휆+휆휔 = 휆(1+휔) = 휆휔 , we can see that 휙2 is deﬁned as 휙2(푎+휆푏) = 푎+휆(푏휔). In a similar fashion, all of the automorphisms of system V can be found:

2 휙1(푎 + 휆푏) = 푎 + 휆푏 휙2(푎 + 휆푏) = 푎 + 휆(푏휔) 휙3(푎 + 휆푏) = 푎 + 휆(푏휔 ) 2 2 2 2 2 2 2 휙4(푎 + 휆푏) = 푎 + 휆(푏 ) 휙5(푎 + 휆푏) = 푎 + 휆(푏 휔) 휙6(푎 + 휆푏) = 푎 + 휆(푏 휔 ) 54 5.4 Automorphisms of Sandler’s Construction of order 332

This is a speciﬁc case of a general construction discovered by Sandler, [20]. In

2 this case, 푆 consists of elements of the form 푎0 + 휆푎1 + 휆 푎2, where 푎푖 ∈ 픽27, with addition deﬁned in the standard way and multiplication deﬁned by

(휆(푖푥) ∗ (휆(푗푦) = 휆(푖+푗푥3푗 푦

휆(3 = 휔

휆(4 = 휆휔

3 where 휔 is a primitive element of 픽27 satisfying 휔 + 2휔 + 1 = 0. This semiﬁeld will be 9 dimensional over 픽3, and an obvious choice of basis would be

풜 = {1, 휔, 휔2, 휆, 휆휔, 휆휔2, 휆2, 휆2휔, 휆2휔2}

We will let 퐴 denote the Knuth cube deﬁned by this basis. Note that |퐶(9, 3)| ≈

1.3 × 1034, so testing each 퐶 ∈ 퐶(9, 3) would be unfeasible even with the aid of a

computer. On the other hand |푆| = 19, 683, so we can use the results from section 5.2

to ﬁnd the automorphisms of 푆. First we determine whether 푆 has any left primitive elements. To do this we check the matrix for left multiplication by each element of

푆 with respect to ℬ and ﬁnd its characteristic polynomial. By Theorem 4.21 such

a polynomial will be primitive of degree 9 over 픽3. Let 푅 be the set of all 푧 ∈ 푆 for which the characteristic polynomial of the matrix for left multiplication by 푧 is

푥9 + 2푥6 + 푥2 + 2푥 + 1. By computation, |푅| = 39, and one of the elements in

푅 is 2휔2 + 휆. Let 푃 be the matrix whose 푖-th row is the vector corresponding to

2 (푖−1 −푇 (2휔 + 휆) , and let 퐵 = [푃 , 푃 , 푃 ] × 퐴. Note that 퐵 is generated by ℒ(2휔2+휆),

and any other basis which generates 퐵 must equal ℒ푧 for some 푧 ∈ 푅. Thus, to −푇 ﬁnd 푄 such that 퐵 = [푄, 푄, 푄 ] × 퐵, we need only consider matrices where 푖-th row is the vector corresponding to the 푖-th term of ℒ푧 with respect to ℒ(2휔2+휆). 55 By computation, 13 such matrices were found; hence there are 13 automorphisms.

For each such 푄, we can determine the automorphism with respect to 풜 by looking −1 at 푃 푄푃 . Since |Aut(푆)| = 13, we know Aut(푆) ≅ ℤ13, and we thus only need to determine one non-trivial automorphism to generate the group. We make the −1 following choice for 푃 푄푃 :

⎛ 1 0 0 0 0 0 0 0 0 ⎞ ⎜ ⎟ ⎜ 0 1 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 1 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 0 2 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 2 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 1 2 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 0 2 1 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 2 1 2 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 0 1 1 1 ⎠

2 Let 푧 ∈ 푆 have the form 푧0 + 휆푧1 + 휆 푧2. Then the automorphisms of 푆 will clearly 2 preserve 푧0 and act on 휆푧1 and 휆 푧2 independently. Consider 휙(휆푧1), with 푧1 = 2 푏0 + 푏1휔 + 푏2휔 where 푏푖 ∈ 픽3. The center block of the matrix then tells us 휙(휆푧1) = 2 휆(푏2 + (2푏0 + 2푏2)휔 + 2푏1휔 . Note that

2 2 3 2휔(푏0 + 푏1휔 + 푏2휔 ) = 2푏0휔 + 2푏1휔 + 2푏2휔 2 = 2푏0휔 + 2푏1휔 + 2푏2(2 + 휔) 2 = 푏2 + (2푏0 + 2푏2)휔 + 2푏1휔

2 2 2 Thus 휙(휆푧1) = 휆(2푧1휔). Similarly, 휙(휆 푧2) = 휆 (2푧2휔 + 푧2휔 ). Since 휔 is primitive, we can deﬁne these mappings as powers of 휔 to get

2 14 2 4 휙(푧0 + 휆푧1 + 휆 푧2) = 푧0 + 휆(푧1휔 ) + 휆 (푧2휔 )

56 As previously noted, this automorphism generates the automorphism group, and

therefore any automorphism of this semiﬁeld will have the form

2 14푖 2 4푖 휙푖(푧0 + 휆푧1 + 휆 푧2) = 푧0 + 휆(푧1휔 ) + 휆 (푧2휔 ) for some positive integer 1 ≤ 푖 ≤ 13, with the identity automorphism corresponding to 푖 = 13.

57 Chapter 6

Constructions and Enumerations

In this chapter we address the enumeration and construction of semifields. We will start by discussing the orders for which all semifields have been found (enumerated), and then give a brief description of the classical algebraic constructions. The methods used to enumerate semifields are all equivalent to determining Knuth cubes for all such semifields, but this yields little information about the algebraic structure of those semifields. For this reason, when we discuss the algebraic constructions we will also investigate what form a Knuth cube corresponding to each construction would have. Knowing this we can then determine which constructions corresponds to a semifield with a given set of Knuth cubes.

6.1 Early Enumeration - Kleinfeld and Walker

As mentioned in chapter 2, the smallest proper semifields are of order 16. It is only fitting then that these were the first semifields to be completely determined. In

1960, Kleinfeld, [10], determined all 16-element Veblen-Wedderburn systems, which satisfy all of the semiﬁeld axioms except for left distributivity, i.e. 푎 ∗ (푏 + 푐) was

not assumed to equal 푎 ∗ 푏 + 푎 ∗ 푐. The lack of zero divisors causes the multiplication table for the nonzero elements of a Veblen-Wedderburn system to be a Latin square.

Kleinfeld used this fact and some linear algebra to show that the entire multiplica-

tion table could be determined once a small number of entries were determined. He

then used a computer to generate and sort all of the possible multiplication tables for

58 Veblen-Wedderburn systems. Since semifields are a special case of these systems, this list includes all of the 16-element semifields as well. There are 24 such semifields, up to isomorphism, in three isotopism classes. The first class consists only of 픽16. The second consists of system W and 4 other semifields. The third consists of system V and 17 other semifields.

Shortly after this, in 1963, Walker, [21], determined all of the 32-element semi-

fields. In [3], Albert proved that any 32-element semifield is isotopic to a 32-element semifield which has a right cyclic basis ℬ = {1, 푏, 푏2, 푏2 ∗푏, (푏2 ∗푏)∗푏}, where (푏2 ∗푏)∗푏 was either equal to 푏2+1 or 푏+1. Walker noted that the multiplication in the semifield is entirely determined by the products of the elements of ℬ, so he used a computer to determine all of the possible products of the basis elements. Note that this is essentially equivalent to finding all of the cubical arrays determined by such a basis.

Walker then computed all of the possible isotopes of these semiﬁelds using a result similar to Theorem 2.13. Walker discovered that there were 6 isotopism classes for the 32 element semiﬁelds, and provided a representative from each class.

6.2 Recent Enumerations by Rúa, Combarro, and Ranilla

For about 45 years after Walker’s work no further enumerations were attempted, which is likely due to a lack of computing power. The semifields of order 27 could likely have been computed, but there was no need due to Menichetti, [13], proving that all semifields of dimension 3 are Albert twisted fields (see section 6.12).

By Theorem 3.4, every semiﬁeld of order 푝푛 is deﬁned by an 푛-dimensional

Knuth cube over 픽푝. Suppose we wanted to enumerate the semiﬁelds of order 64. We would consider all cubical arrays, 퐴, of dimension 6 over 픽2 which are in standard form. Then 퐴1∗∗ = 퐴∗1∗ = 퐼, and 퐴푖푗∗ ≠ (0, 0, 0, 0, 0, 0) otherwise 퐴 would not be 6 nonsingular. Thus, for 1 < 푖, 푗 ≤ 6, 퐴푖푗∗ has 2 − 1 = 63 possible values. This results 59 in 6325 ≈ 9.6 × 1044 possible choices for 퐴 which would each then need to be tested for nonsingularity. This would be a straightforward method which would take far

too long to be practical. And, as the dimension of the semiﬁelds increases, so too

does the number of cubical arrays which would need to be tested. For the 81-element

semifields only 809 ≈ 1.3 × 1017 would need to be tested, while for the 243-element semifields, 24216 ≈ 1.3 × 1038 would need to be tested. Thus the developments in enumerating semifields based on cubical arrays are based on finding ways to reduce the number which need to be tested. Results such as that used by Walker can help quite a bit, as simply knowing that only two possible cyclic bases exist reduces the number of cubical arrays from 3116 to 3111. This number can be further reduced by building the cubical arrays to be nonsingular. Though it may seem like it would be slower, it is actually far quicker to make successive choices of 퐴푖푗∗ by removing linear combinations of previous choices. For example, in a 4- dimensional cubical array, if 퐴22∗ = (0, 1, 1, 0), then 퐴2푗∗ and 퐴푖2∗ cannot equal any

scalar multiple (0, 1, 1, 0). Similarly, 퐴42∗ cannot be a linear combination of 퐴12∗,

퐴22∗ and 퐴32∗. This is essentially the approach taken by Rúa, Combarro, and Ranilla in [18], which enumerates all of the 64 element semifields and was published in 2009. Since then, they have continued to refine their search algorithm to enumerate the semifields of order 35 and 74 in [19] and [4] respectively. It is interesting to note that the problem has shifted more towards the realm of computer science, as the major advancement in [4] was in a more efficient implementation of their search algorithm which allowed for parallel processing.

60 6.3 Reverse Decimal Matrix Notation

In this section we introduce a useful notation for discussing the properties of

Knuth cubes. For a cubical array, 퐴, of dimension 푛 over 픽푝, we will construct a matrix, 퐴, where 퐴푖푗 is a number representing the vector 퐴푖푗∗. To ensure that this notation is well-deﬁned, we will use a variant of changing from base 푝 to base 10.

푛 ̂0 ̂ First, let us identify the standard basis vectors of 픽푝 as follows: 푝 = 1 = 푒1,

̂푛−1 ̂푝= 푒2, ⋯, 푝 = 푒푛. Then, deﬁne the sums of vectors (modulo 푝) as

푛 푛 푛 ̂ ̂푘−1 푘−1 (6.1) ∑ 푎푘푝 = ∑ 푎푘푒푘 = ∑ 푎푘푝 푘=1 푘=1 푘=1

푛 Thus, given a vector 푣 ∈ 픽푝 , we treat the coordinates of 푣 as the reversed digits of a number in base 푝. By reversing these digits and converting the number to base

4 10, we get the reverse decimal form of 푣.For example, in 픽2, the basis vectors are 1̂ = (1, 0, 0, 0), 2̂ = (0, 1, 0, 0), 4̂ = (0, 0, 1, 0) and 8̂ = (0, 0, 0, 1). Addition is modulo

2, so 2̂ + 2̂ = 0. To ﬁnd the reverse decimal form of the vector (0, 1, 1, 1) we would consider the base 10 value of the binary number 1110, which is 14, so the reverse decimal form of (0, 1, 0, 1) is 14̂. Alternatively, we could use the deﬁnition of the standard basis and equation 6.1 to get

̂ ̂ ̂ ̂ (0, 1, 1, 1) = 푒2 + 푒3 + 푒4 = 2 + 4 + 8 = 14

Next, we apply this to a cubical array by converting the vectors 퐴푖푗∗. That is, given a cubical array 퐴, we deﬁne the matrix 퐴 as

푛 ̂푘−1 (6.2) 퐴푖푗 = ∑ 푝 퐴푖푗푘 푘=1

We will call 퐴 the reverse decimal matrix of 퐴, and denote their relationship by

퐴 ∼ 퐴. Similarly, if ̂푣 is the reverse decimal form of 푣, then we will write 푣 ∼. ̂푣 The hat notation is used in this section to emphasize that the digits used are representing 61 vectors and not numbers. The hat is not necessary once the notation is understood, and for this reason the results in the following sections will not have it.

This notation has many beneﬁts. First, it is easier to present than a cubical array, but can easily be converted into a cubical array using (6.2). Second, the matrices for left and right multiplication by various basis elements can be computed.

To find the matrix 푅 for right multiplication by 푒푗, consider the column 퐴∗푗. Then for each 퐴푖푗 in that column, and each 푣 associated with it by 6.1, 푅푖∗ = 푣. Similarly, to find the matrix 퐿 for left multiplication by 푒푖, consider the row 퐴푖∗. Then, for each 퐴푖푗 in that row, and each 푣 corresponding to it, 퐿푗∗ = 푣. A third benefit of the reverse decimal matrix is that it can easily be used as a multiplication table for the basis elements of 푆. Given basis elements 푝̂푎, 푝̂푏, with

푎 ̂푏 0 ≤ 푎, 푏 < 푛, the product 푝̂ ⋅ 푝 is 퐴(푎−1),(푏−1). Since the first row and column of 퐴 represent multiplication by the identity element, to find 푝̂푎 ⋅ 푝̂푏 one simply finds the intersection of the row starting with 푝̂푎 and the column starting with 푝̂푏. Finally, the following result regarding cubical arrays becomes far simpler when it is used with reverse decimal matrices.

′ Theorem 6.1. A semifield, 푆, of dimension 푛 over 픽푝 has a subsemifield, 푆 , of dimension 푚 over 픽푝 if and only if there exists 퐴 ∈ 퐾(푆) such that the cubical array ′ 퐴 consisting of the first 푚 × 푚 × 푚 entries of 퐴 is nonsingular, and 퐴푖푗푘 = 0 for all 푖, 푗 ≤ 푚, 푘 > 푚.

Proof. First suppose 푆 has a subsemiﬁeld 푆′ of dimension 푚. Then there exists a

′ ′ basis {1 = 푥1, 푥2, ..., 푥푚} of 푆 over 픽푝 which generates a Knuth cube 퐴 . Further, 푆

62 must have a basis of the form {1, 푥2, ..., 푥푚, 푦푚+1, ..., 푦푛−1} which generates a Knuth cube 퐴 ∈ 퐾(푆). Then we have

푛 푚 푛 푚 ′ ′ 푥푖 ∗ 푥푗 = ∑ 퐴푖푗푘푥푘 = ∑ 퐴푖푗푘푥푘 + ∑ 퐴푖푗푘푥푘 = ∑ 퐴푖푗푘 푘=1 푘=1 푘=푚+1 푘=1

Thus 퐴푖푗푘 = 0 when 푖, 푗 ≤ 푚 and 푘 > 푚. Now suppose the converse is true, that there exists 퐴 ∈ 퐾(푆) such that the cubical array 퐴′ consisting of the ﬁrst 푚 × 푚 × 푚 entries of 퐴 is nonsingular, and

′ 퐴푖푗푘 = 0 for all 푖, 푗 ≤ 푚, 푘 > 푚. Note that 퐴 will clearly be in standard form as ′ well. Thus, if {1 = 푥1, 푥2, ..., 푥푚} is the basis which generates 퐴 , then it is also the basis of a semiﬁeld of dimension 푚 over 픽푝, where multiplication is deﬁned by

푛 ′ 푥푖 ∗ 푥푗 = ∑ 퐴푖푗푘푥푘 푘=1

′ Corollary 6.2. A semiﬁeld, 푆, of dimension 푛 over 픽푝 has a subsemiﬁeld, 푆 , of

dimension 푚 over 픽푝 if and only if there exists a reverse decimal matrix 퐴 for 푆 푚 where 퐴푖푗 < 푝 for all 1 ≤ 푖, 푗 ≤ 푚

63 6.4 Examples

To further illustrate the use of the reverse decimal matrix, consider the reverse

decimal matrices associated with the Knuth cubes already given for 픽16, system V, and system W. ⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ 2 3 8 12 ⎟ ⎜ ⎟ 픽16 퐹 ∼ ⎜ ⎟ ⎜ 4 8 6 11 ⎟ ⎜ ⎟ ⎝ 8 12 11 13 ⎠

⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ 2 2 12 4 ⎟ System W 푊 ∼ ⎜ ⎟ ⎜ 4 8 2 3 ⎟ ⎜ ⎟ ⎝ 8 12 1 2 ⎠

⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ 2 3 12 4 ⎟ System V 푉 ∼ ⎜ ⎟ ⎜ 4 8 5 14 ⎟ ⎜ ⎟ ⎝ 8 12 15 9 ⎠

This notation makes a few things very clear. Notice that the matrix for 픽16 is sym- metric, which is due to the commutativity of its multiplication. Also notice that the

upper left 2 × 2 entries of these matrices are all less than 4. This means that each of these semifields contains a subsemifield of dimension 2 over 픽2. Since there are no proper 4-element semifields, this means that each of these subsemifields is isomorphic to 픽4. The reason for this is due to the fact that in all three of these semifields

(푎+휆0)∗(푐+휆0) = 푎푐, and 푎, 푐 ∈ 픽4. Furthermore, the fact that the second columns

64 of these matrices are all equal is due to the fact that (0 + 휆푏) ∗ (푐 + 휆0) = 휆(푏푐) in all three of these semiﬁelds.

6.5 Knuth Cubes and Constructions

Suppose we have a Knuth cube 퐴 of dimension 푛 over 픽푝, and suppose we have a semiﬁeld 푆 of order 푝푛 generated by some algebraic construction. Then 퐴 will correspond to 푆 if and only if 퐴 ∈ 퐾(푆). Thus, we construct a Knuth cube 퐵 for 푆, and, using 퐶(푛, 푝) or the results from section 5.2, we can generate 퐾(푆) and see if 퐴 is in 퐾(푆). Alternatively, we could use 퐴 to construct a semiﬁeld 푆′ and investigate all of the cubes in 퐾(푆′) to see if any of them correspond to a cube for a given construction.

The key to constructing a Knuth cube for a semiﬁeld 푆 generated by a given construction is the determination of an appropriate basis to generate the cube. Since

퐾(푆) will contain the Knuth cubes generated by all appropriate bases, we can choose a basis of the most convenient form. For some constructions, such as those used for system V and system W, there is an obvious choice of basis. But there are constructions where it is not clear what a valid basis would be. Thus, most of the work in the following sections is devoted to determining an appropriate basis for a construction. Once a basis is found, general products of the basis elements are determined, and the general form of a reverse decimal matrix generated by this basis is given.

6.6 Albert Binary Semiﬁelds

According to Wene, [25], this construction came from an investigation by Albert of the Knuth binary semiﬁelds described in the following section. We introduce it

65 ﬁrst due to it’s straightforward construction.

푛 Consider the vector space 푈 = 픽2 , for an odd 푛 ≥ 3. Let 푈 = 푉 + 휔픽2, with

1 ∈ 푉 . In other words, if 푈 has a basis {푥1, ..., 푥푛} and 휔 = 푥푖, then 푉 is the span

of {푥1, ..., 푥푖−1, 푥푖+1, ..., 푥푛}. Then we construct a semiﬁeld 푆 = (푈, +, ∗) where, for 푎, 푏 ∈ 푉 , 푎 ∗ 푏 = 푎푏

푎 ∗ 휔 = 휔 ∗ 푎 = 푎휔 + 푎2 + 푎

휔 ∗ 휔 = 휔2 + 1 Due to the many possible choices of 휔, it is not clear what form a basis for an Albert

binary semiﬁeld would have. However, in the case where 푛 = 2푘 +1, we will be able to

푛 determine such a basis. Wene, [25], proved that 휔 is a defining element of 픽2 over 픽2, 푛 i.e. 푈 = 픽2[휔], and that if 휎 is an automorphism of 픽2 , then 휎(푈) = 휎(푉 ) + 휎(휔)픽2 will define an isomorphic semifield. This leads to the following lemma.

Lemma 6.3. Let 푆 be an Albert binary semiﬁeld of order 2푛, where 푛 = 2푘+1 for some

푘. Then 푆 is isomorphic to an Albert binary semiﬁeld with a basis {1, 훼, 훼2, ..., 훼푛−1 =

푛 휔}, where 훼 is a deﬁning element of 픽2 over 픽2,.

푛 푛−1 2푘 Proof. Let 훼 be a deﬁning element of 픽2 . By assumption, 훼 = 훼 , and clearly 푘 < 푛 − 1. Thus 훼푛−1 is a conjugate of 훼. By Theorem 2.21 in [12], there is an

automorphism which maps 훼 to 훼2푘 . Thus, 푆 is isomorphic to a semiﬁeld with a

basis {1, 훼, ..., 훼푛−1 = 휔}.

By Theorem 3.33 in [12], if 푚(푥) = 푥푛 + ∑푛−1 푎 푥푖 is the minimal polynomial 푖=0 푖 of 훼, then it is also the minimal polynomial of 훼2푘 , and it is irreducible. Thus the

number of possible Albert binary semiﬁelds of order 22푘+1 is less than or equal to the

푘 number of irreducible polynomials of order 2 + 1 over 픽2. Now consider the reverse decimal matrices 퐴 for an Albert binary semiﬁeld with

66 푛−1 푛 푛−1 basis {1, 훼, ..., 훼 = 휔}, and 퐵 for 픽2 with basis {1, 훼, ..., 훼 }. Then 퐴푖푗 = 퐵푖푗 for 1 ≤ 푖, 푗 < 푛, since these entries correspond to multiplication in the ﬁeld.

To determine whether a given (2푘 + 1) × (2푘 + 1) reverse decimal matrix 퐵

defines an Albert binary semifield, check that the first 2푘 × 2푘 entries match a reverse

2푘+1 2푘+1 decimal matrix 퐴 for 픽2 . Then, consider 푎 = 퐴3,2푘 , and 푎 ∈ 픽2 such that

푎 ∼ 푎. Then, the minimal polynomial of 훼 over 픽2 will be

2푘+1 2푘 푚(푥) = 푥 + 푎2푘 푥 + ⋯ + 푎1푥 + 푎0

This can then be used to determine whether the last column of 퐵 satisﬁes the structure of an Albert binary semiﬁeld.

6.7 Knuth Binary Semiﬁelds

This construction was introduced by Knuth in [11], and is called “binary” due to the fact that the semiﬁelds resulting from this construction are vector spaces over

픽2. 푛−1 푚푛 푚 Let {1, 훼, ..., 훼 } be a basis for 픽2 over 픽2 , where 푛 is odd and 푛푚 > 3. 푚푛 푚 푛−1 푖 Let 푓 ∶ 픽2 ↦ 픽2 be a linear function deﬁned by 푓(훼 ) = 1 and 푓(훼 ) = 0 for 푚푛 0 ≤ 푖 ≤ 푛 − 2. Then, deﬁne two products ∘ and ∗ on 픽2

푎 ∘ 푏 = 푎푏 + (푓(푎)푏 + 푓(푏)푎)2

푎 ∘ 푏 = (1 ∘ 푎) ∗ (1 ∘ 푏)

67 푚푛 푛푚 Then (픽2 , +, ∘) is a presemiﬁeld (it has no multiplicative identity) and (픽2 , +, ∗) is a semiﬁeld.

To ﬁnd a basis, we start with the case where 푚 = 1. This gives us

1 ∘ 1 = 1

1 ∘ 훼푖 = 훼푖 0 ≤ 푖 ≤ 푛 − 2

1 ∘ 훼푛−1 = 훼푛−1 + 1

훼푖 ∘ 훼푗 = 훼푖+푗 0 ≤ 푖, 푗 ≤ 푛 − 2

훼푖 ∘ 훼푛−1 = 훼푖+푛−1 + 훼2푖 0 ≤ 푖 ≤ 푛 − 2

훼푛−1 ∘ 훼푛−1 = 훼2푛−2

1 ∘ (훼푛−1 + 1) = 훼푛−1 This deﬁnes the following products, for 0 ≤ 푖, 푗 ≤ 푛 − 2

1 ∗ 1 = (1 ∘ 1) ∗ (1 ∘ 1) = 1

1 ∗ 훼푖 = (1 ∘ 1) ∗ (1 ∘ 훼푖) = 훼푖

1 ∗ 훼푛−1 = (1 ∘ 1) ∗ (1 ∘ (훼푛−1 + 1)) = 훼푛−1

훼푖 ∗ 훼푗 = (1 ∘ 훼푖) ∗ (1 ∘ 훼푗) = 훼푖+푗

훼푖 ∗ 훼푛−1 = (1 ∘ 훼푖) ∗ (1 ∘ (훼푛−1 + 1)) = 훼푖(훼푛−1 + 훼푖 + 1)

훼푛−1 ∗ 훼푛−1 = (1 ∘ (훼푛−1 + 1)) ∗ (1 ∘ (훼푛−1 + 1)) = 훼2푛−2 + 1 More simply 1 ∗ 1 = 1

1 ∗ 훼푖 = 훼푖

1 ∗ 훼푛−1 = 훼푛−1

훼푖 ∗ 훼푗 = 훼푖+푗

훼푖 ∗ 훼푛−1 = 훼푖훼푛−1 + (훼푖)2 + 훼푖

훼푛−1 ∗ 훼푛−1 = (훼푛−1)2 + 1 This is exactly the case of an Albert binary semiﬁeld with 휔 = 훼푛−1. Now suppose

푚−1 푚 푖 푛−1 푖 푚 > 1. Let {1, 훽, ..., 훽 } be a basis for 픽2 . Then 푓(훽 훼 ) = 훽 . This causes 68 some slight changes to the multiplication, giving the more general results, for 0 ≤

푘 ≤ 푚 − 1

훽푏 ∗ 훽푘 = 훽푏+푘

훽푏 ∗ 훽푘훼푖 = 훽푏+푘훼푖

훽푏 ∗ 훽푘훼푛−1 = 훽푏+푘훼푛−1

훽푏훼푖 ∗ 훽푘훼푗 = 훽푏+푘훼푖+푗

훽푏훼푖 ∗ 훽푘훼푛−1 = (훽푏+푘훼푖)훼푛−1 + (훽푏+푘훼푖)2 + 훽푏+푘훼푖

훽푏훼푛−1 ∗ 훽푘훼푛−1 = 훽푏+푘(훼푛−1)2 + 훽푏+푘

This is similar to the case of the Albert binary semiﬁelds where 휔 = 훼푛−1. In this case, there is a basis of the form

{1, 훽, ..., 훽푚−1, 훼, ..., 훽푚−1훼, ..., 훼푛−1, ..., 훽푚−1훼푚−1}

If 퐵 is the reverse decimal matrix for 푆 generated by this basis, then there is a reverse

푚푛 decimal matrix 퐴 for 픽2 such that 퐵푖푗 = 퐴푖푗 for all 1 ≤ 푖, 푗 ≤ 푚(푛 − 1). To ﬁnd 푛 퐵푖푗 where 푖, 푗 > 푚(푛 − 1), we need to determine the value of 훼 . This can be done using the fact that the reverse decimal form of 훼푛 is located in the 푚(푛 − 1) + 1

column and 푚 + 1 row. Once that is determined, the remaining entries of 퐵 can be checked.

6.8 Sandler’s Construction

A Sandler semiﬁeld, 푆, has 푝푛푚2 elements, where 푚 > 1, of the form

2 푚−1 푛푚 푎0 + 휆푎1 + 휆 푎2 + ⋯ + 휆 푎푚−1; 푎푖 ∈ 픽푝

Multiplication is deﬁned by

(휆(푖푥)(휆(푗푦) = 휆(푖+푗푥(푝푛)푗 푦

휆(푚 = 훿 69 푛 where 훿 is a root of an irreducible polynomial of degree 푚 over 픽푝 , and the exponential 푛푚 notation refers to repeated left multiplication. Let 픽푝 = 픽푝[훼] for some 훼. We can now construct a reverse decimal matrix, 퐵 for 푆 in a straightforward manner. First, we choose a basis of the form

{1, 훼, ..., 훼푛푚−1, 휆, 휆훼, ..., 휆훼푛푚−1, ..., 휆(푚−1, 휆(푚−1훼, ..., 휆(푚−1훼푛푚−1}

Then 퐵 will have a block structure corresponding to which power of 휆 is multiplied

푛푚 by the basis of 픽푝 . The notation 퐵[푥,푦] will denote the 푛푚 × 푛푚 submatrix whose

upper-left corner is 퐵푥푚+1,푦푚+1. Here’s a graphic to help visualize 퐵

1 휆 ⋯ 휆(푚−1

1 ⎛ ⎞ ⎜ ⎟ 휆 ⎜ ⎟ ⎜ ⎟ ⋮ ⎜ ⎟ ⎜ ⎟ (푚−1 휆 ⎝ ⎠ 푛푚 푛푚−1 Then 퐵[1,1] will be the reverse decimal matrix for 픽푝 generated by the basis {1, 훼, ..., 훼 }. (푖 (푖+1)푛푚 Similarly, the entries in 퐵[푖,1] will equal the entries of 퐵[1,1] times 휆 ∼ 푝 . The rest of 퐵 is less straightforward and depends on the powers of 훼.

To tell if a particular 푛푚2 × 푛푚2 reverse decimal matrix corresponds to a San-

dler semiﬁeld, ﬁrst determine whether 퐵[1,1] corresponds to a reverse decimal matrix 푛푚 (푖+1)푚 for 픽푝 , and then check whether 퐵[푖,1] will equal the entries of 퐵[1,1] times 푝 .

From there, determine which basis generated 퐵[1,1] and then check the remaining blocks of 퐵.

6.9 Petit’s Construction

This construction was introduced by Petit, [15], [16], and is based on factor

rings of skew polynomial rings over finite fields. We do not give a formal definition 70 of these rings or their properties, but instead simply provide sufficient information to

derive the construction.

Let 픽푝푛 [푋; 휎] denote the skew polynomial ring over 픽푝푛 , and 휎 is an automor- 휎 푝푖 phism of 픽푝푛 , i.e. 푎 = 푎 for some 푖. Multiplication is deﬁned as usual on the 푝푖 left, but 푋푎 = 푎 푋. Wene, [25], cites theorems that show that 픽푝푛 [푋; 휎] has a

division algorithm, and thus can be factored by ideals. Let 푀(푋) ∈ 픽푝푛 [푋, 휎] be

an irreducible polynomial of degree 푚, and let 퐹푀 = 픽푝푛 [푋; 휎]/푀(푋). Addition

is deﬁned on 퐹푀 as normal, but multiplication is deﬁned by 퐴(푋) ∗ 퐵(푋) = 푅(푋)

where 퐴(푋)퐵(푋) = 푄(푋)푀(푋)+푅(푋) in 픽푝푛 [푋; 휎]. Then (퐹푀 , +, ∗) is a semiﬁeld of order 푝푛푚 .

푚−1 Note that {1, 푋, ..., 푋 } forms a basis of 퐹푀 over 픽푝푛 , and if 푛 > 1, and {1, 훼, ..., 훼푛−1} is a basis of 퐹 , then a basis for 푆 over 퐹 would be

{1, 훼, ..., 훼푛−1, 푋, ..., 훼푛−1푋, ..., 푋푚−1, ..., 훼푛−1푋푚−1}

Let 퐵 be a reverse decimal matrix for 푆 with respect to this basis. Then there is a reverse decimal matrix 퐴 for 픽푝푛푚 such that 퐵푖∗ = 퐴푖∗ for 1 ≤ 푖 ≤ 푚 − 1. From there, it should be easy to determine if there exists a ﬁxed 휎 such that 푋 ⋅ 푎 = 푎휎푋.

6.10 Dickson’s Even-Dimensional Semiﬁelds

The study of ﬁnite semiﬁelds began with the work of Dickson in the early 1900s.

This particular construction was introduced in [5].

Let 푝 be odd, 푛 > 1, and let 푓 ∈ 픽푝푚 be a nonsquare element. Then, the

Dickson semiﬁeld 푆 is a two dimension vector space over 픽푝푛 with basis {1, 휆}, and multiplication deﬁned by

(푎 + 휆푏)(푐 + 휆푑) = (푎푐 + 푓(푏푑)휃) + 휆(푎푑 + 푏푐)

71 휃 푝푟 where 휃 is an automorphism of 픽푝푛 , i.e. 푥 = 푥 for some 1 ≤ 푟 < 푛. 푛−1 푛 푛−1 푛−1 Let {1, 훼, ..., 훼 } be a basis of 픽푝 . Then 푆 has a basis of the form {1, ..., 훼 , 휆, ..., 휆훼 }, and its corresponding reverse decimal matrix will have the form

푛−1 푛 2푛−1 ⎛ 1 푝 ⋯ 푝 푝 ⋯ 푝 ⎞ ⎜ ⎟ ⎜ 푝 ⎟ ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ 푛−1 ⎟ 퐴 = ⎜ 푝 ⎟ ⎜ ⎟ ⎜ 푝푛 ⎟ ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎟ 2푛−1 ⎝ 푝 ⎠

We will let 퐴[푥,푦] denote the 푛×푛 submatrix of 퐴 whose upper left corner is located at 푛 퐴(푛푥,푛푦). Then 퐴[1,1] corresponds to a reverse decimal matrix for 픽푝 , 퐴[2,1] = 퐴[1,2] = 푛 2 푝 퐴[1,1]. The entries in 퐴[2,2] require a bit more work. First, note that 퐴푛푛 ∼ 휆 = 푓, 휃 and 퐴(푛,푛+1) = 퐴(푛+1,푛) ∼ 푓(훼) . Thus, it is possible to determine what 푓 and 휃 must be, which can then be used to determine whether this is a Dickson semiﬁeld.

6.11 Quadratic Over a Weak Nucleus

This construction was given by Knuth in [11], and both system V and system

W are special cases of this construction. A weak nucleus is a subset 푁 of a semiﬁeld 푆

such that, for 푎, 푏, 푐 ∈ 푆, 푎∗(푏∗푐) = (푎∗푏)∗푐 is true whenever any two of the elements

is in 푁. Knuth constructs 푆 as a two-dimensional vector space over a ﬁnite ﬁeld 픽푝푛 ,

and assumes that 픽푝푛 will be a weak nucleus of 푆. The elements of 푆 have the form

(푎 + 휆푏), for 푎, 푏 ∈ 픽푝푛 . Knuth proves that this forces (0 + 휆푏) ∗ (푐 + 휆0) = 휆(푏푐),

72 휎 휎 푝푖 and (푎 + 휆0) ∗ (0 + 휆푑) = 휆(푎 푑) where 휎 is an automorphism of 픽푝푛 , i.e. 푎 = 푎 for 1 ≤ 푖 ≤ 푛. From here we have the general product

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 휆(푏푐 + 푎휎푑) + (휆푏) ∗ (휆푑)

Knuth then considers the results for when 푝 ≠ 2 or 푝 = 2, but we will only present the results of this investigation here. In both cases, 푆 will be a two dimensional vector

푛−1 푛−1 푛−1 space over 픽푝푛 , so, if {1, 푥, ..., 푥 } is a basis of 픽푝푛 , then {1, ..., 푥 , 휆, ..., 휆푥 } 2 will be a basis for 푆, such that 휆 ∉ 픽푝푛 . Let 퐴 be the reverse decimal matrix for this

basis, and let 퐴[푥,푦] denote the 푛 × 푛 submatrix whose upper left order is at 퐴(푛푥,푛푦).

Then 퐴[1,1] will be a reverse decimal matrix for 픽푝푛 .

In the ﬁrst case, suppose 푝 ≠ 2, 훼, 훽, 휎 ∈ Aut(픽푝푛 not all trivial, 푓 ∈ 픽푝푛 nonsquare, multiplication in 푆 is deﬁned by

(푎 + 휆푏)(푐 + 휆푑) = (푎푐 + 푏훼푑훽푓) + 휆(푎휎푑 + 푏푐)

푛 휎 Then 퐴[2,1] = 푝 퐴[1,1]. To determine 휎, note that 퐴(2,푛+1) ∼ (푥)(휆) = 휆(푥 ), and 휎

will determine all of 퐴[1,2]. For 퐴[2,2], note the following

2 퐴(푛+1,푛+1) ∼ 휆 = 푓 훽 퐴(푛+1,푛+2) ∼ 휆(휆푥) = 푥 푓 훼 퐴(푛+2,푛+1) ∼ (휆푥)휆 = 푥 퐹

These can be used to determine 푓, 훼, and 휎, and then verify the rest of 퐴.

−1 Now suppose 푝 = 2. Let 휎 ∈ Aut(픽푝푛 ) be nontrivial, 휏 = 휎 , and let 푓, 푔 ∈ 픽푝푛 such that

휎+1 푦 + 푔푦 − 푓 ≠ 0; for all 푦 ∈ 픽푝푛

73 Then the multiplication (푎 + 휆푏)(푐 + 휆푑) has one of four forms

(1) (푎푐 + 푏휎푑휏2 푓) + 휆(푏푐 + 푎휎푑 + 푏휎푑휏 푔)

(2) (푎푐 + 푏휎푑푓) + 휆(푏푐 + 푎휎푑 + 푏휎푑푔)

(3) (푎푐 + 푏휏 푑휏2 푓) + 휆(푏푐 + 푎휎푑 + 푏푑휏 푔)

(4) (푎푐 + 푏휏 푑푓) + 휆(푏푐 + 푎휎푑 + 푏푑푔)

Rather than look at each of these four separately, We will describe the general method

for determination. The entries in 퐴[2,1] correspond to products of the form (휆푏)(푐) = 푛 휆(푏푐), so 퐴[2,1] = 푝 퐴[2,1]. The entries in 퐴[1,2] correspond to products of the form 휎 휎 (푎)(휆푑) = 휆(푎 푑). In particular, 퐴(2,푛+1) ∼ (푥)(휆) = 휆(푥 ). From this it is possible to determine 휎 and 휏, and consequently all of 퐴[1,2]. Determining which of the four 2 types the semiﬁeld has depends on 퐴[2,2]. Let 훼1, 훼2, 훽1, 훽2 ∈ {1, 휎, 휏, 휏 }. Then the following can be used to determine 푓, 훼푖, 훽푖:

퐴(푛+1,푛+1) ∼ (휆)(휆) = 푓 + 휆푔

훽1 훽2 퐴(푛+1,푛+2) ∼ (휆)(휆푥) = 푥 푓 + 휆(푥 푔)

훼1 훼2 퐴(푛+2,푛+1) ∼ (휆푥)(휆) = 푥 푓 + 휆(푥 푔)

6.12 Albert’s Twisted Fields

This construction was introduced by Albert in [2], and gets its name due to the

fact that it defines a new product on the elements of a finite field by “twisting” the

existing ﬁnite ﬁeld product.

Let 푛 > 2 and 휎, 휃 ∈ Aut(픽푝푛 . Let 퐹휎 and 퐹휃 denote the subﬁelds of 픽푝푛 which

are ﬁxed by 휎 and 휃, and let 퐹 = 퐹휎 ∩ 퐹휃. Let 푐 ∈ 퐹 such that 푥푦 푐 ≠ 푥휃푦휎 for any nonzero 푥, 푦 ∈ 퐹 . Deﬁne linear transformations 퐿 and 푅 over 퐹 by

퐿(푥) = 푥 − 푐푥휃 푅(푦) = 푦 − 푐푦휎 74 The Albert twisted semiﬁeld, 푆 is constructed from 퐹 with standard addition and multiplication ∗ deﬁned by

퐿(푥) ∗ 푅(푦) = 푥푦 − 푐푥휃푦휎

The complexity of this construction makes it difficult to determine what form a Knuth cube would have if generated by this construction. There is one specific case where it is actually somewhat straightforward. Menichetti, [13], proved that every semifield of dimension 3 over a finite field will be an Albert twisted field, and determined a set of structure constants which would correspond to such a semifield. Since the Knuth cubes are equivalent to the set of structure constants, we can easily adapt his results.

If 푆 is three dimensional over a ﬁeld, then it will have a Knuth cube whose reverse decimal matrix has the form

푘−1 푘 2푘−1 2푘 3푘−1 ⎛ 1 푝 ⋯ 푝 푝 ⋯ 푝 푝 ⋯ 푝 ⎞ ⎜ ⎟ ⎜ 푝 ⎟ ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ 푘−1 ⎟ ⎜ 푝 ⎟ ⎜ ⎟ ⎜ 푝푘 ⎟ 퐴 = ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ 2푘−1 ⎟ ⎜ 푝 ⎟ ⎜ ⎟ ⎜ 푝2푘 ⎟ ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎟ 3푘−1 ⎝ 푝 ⎠

푘 (푖−1)푘 where 퐴[1,1] is a reverse decimal matrix for 퐺퐹 (푝) , 퐴[1,푖] = 퐴[푖,1] = 푝 퐴[1,1] for 2푘 푖 = 2, 3, and 퐴[2,2] = 푝 퐴[1,1].

75 Chapter 7

Extending a Subﬁeld of a Semiﬁeld

As mentioned in chapter 6, cubical arrays have been used to ﬁnd all of the semi-

fields of orders 24, 25, 26, 34, 35, and 74. Also, a number of the known constructions yield block reverse decimal matrices where many of the blocks are related to a reverse decimal matrix for a finite field. This helped inspire the results in this chapter, which provide a method for constructing new semifields by using the Knuth cubes of known semifields over 픽푝 to define new semifields over extensions of 픽푝.

7.1 Cubical Arrays Over Extension of 픽푝 The crux of these results is based on the natural embedding of a finite field into any of its extension fields. Because of this, we will use the following notation:

푚 for 푀 ∈ 퐺퐿(푛, 푝), [푀]푝푚 will denote the matrix in 퐺퐿(푛, 푝 ) whose entries are the

natural embedding of the entries of 푀 into 픽푝푚 . 푚 Lemma 7.1. Let 푀 ∈ 퐺퐿(푛, 푝). Then [푀]푝푚 ∈ 퐺퐿(푛, 푝 ).

Proof. Because 푀 is invertible, det(푀) ≠ 0. Let 휎 denote the natural embedding of

픽푝 into 픽푝푚 . Note that 휎 is an injective homomorphism, thus det(푀) = det([푀]푝푚 ),

and thus [푀]푝푚 is nonsingular.

We will use a similar notation for cubical arrays, with [퐴]푝푚 signifying that

we wish to view the entries of 퐴 as elements of 픽푝푚 in the natural way. Unlike with matrices, nonsingularity of cubical arrays is not necessarily preserved under this

embedding.

In the proof of Theorem 3.4, we showed how the properties of a cubical array 76 translated to the algebra which it deﬁned. If the cubical array is in standard form,

then the resulting construction will have a multiplicative identity. If the cubical array

is nonsingular, then the resulting construction will have no zero divisors. These facts

inspire the following result.

푛 Lemma 7.2. Let 푆 be a semiﬁeld of order 푝 and 퐴 ∈ 퐾(푆). Then [퐴]푝푚 deﬁnes an

푛-dimensional semiﬁeld over 픽푝푚 , if and only if [퐴]푝푛 is nonsingular over 픽푝푚 .

Proof. Let 퐼 denote the 푛 × 푛 identity matrix. By deﬁnition, 퐴1∗∗ = 퐴∗1∗ = 퐼, so

[퐴1∗∗]푝푛 = [퐴∗1∗]푝푛 = 퐼 as well. Thus [퐴]푝푚 is in standard form, and [퐴]푝푚 will deﬁne

a semiﬁeld over 픽푝푚 if and only if it is nonsingular.

Let 푆[푚] denote the system deﬁned by 퐴 over 픽푝푚 . By the Lemma above, 푆[푚]

is a semiﬁeld if and only if [퐴]푝푚 is nonsingular. If 푆[푚] is a semiﬁeld, then there exists an 푛푚 × 푛푚 × 푛푚 cubical array for 푆[푚] over 픽푝. In particular, suppose 퐴 is

deﬁned by a basis {1, 훼2, ..., 훼푛}, and let ℬ = {1, 훽2, ..., 훽푚} be a basis for 픽푝푚 over

픽푝. If [퐴]푝푚 is nonsingular, then the cubical array 퐵 deﬁned by the basis

{1, 훽2, ..., 훽푚, 훼2, 훼2훽2, ..., 훼2훽푚, ..., 훼푛, 훼푛훽2, ..., 훼푛훽푚} will also be nonsingular, and we say that 퐵 is the inflation of 퐴 by ℬ. Similarly, we will say 푆[푚] is the 푚-inflation of 푆. Inflation does more than simply allow a cubical array to define multiple semi-

fields. It also provides a nesting structure to semifields over the same prime subfield, as is demonstrated by the following results.

′ 푛 Theorem 7.3. Let 푆 and 푆 be isotopic semifields of order 푝 . Then 푆[푚] is a semifield ′ ′ if and only if 푆[푚] is a semifield. Further, if 푆[푚] is a semifield, it is isotopic to 푆[푚].

77 Proof. Let 퐴 ∈ 퐾(푆), 퐵 ∈ 퐾(푆′). By Theorem 3.16, there exist 퐹 , 퐺, 퐻 ∈ 퐺퐿(푛, 푝) −푇 such that 퐵 = [퐹 , 퐺, 퐻 ] × 퐴 and

푛 푛 푇 푇 퐵푖∗∗ = ∑ 퐹 푖푗퐺퐴푖∗∗퐻 = 퐺 (∑ 퐹 푖푗퐴푖∗∗) 퐻 푗=1 푗=1

푛 Let 푣 ∈ 픽푝푚 and 푤 = 푣[퐹 ]푝푛 . Then, using the equation above and embedding into

픽푝푚 , we get

푛 푛 푛 푇 ∑ 푣 [퐵 ] 푚 = [퐺] 푚 (∑ ∑ 푣 [퐹 ] 푚 [퐴 ] 푚 )[퐻 ] 푚 푖=1 푖 푖∗∗ 푝 푝 푖 푖푗 푝 푗∗∗ 푝 푝 푖=1 푗=1

푛 푇 = [퐺]푝푚 (∑ 푤푗[퐴푗∗∗]푝푚 )[퐻 ]푝푚 푗=1

푇 푛 By Lemma 7.1, [퐺] 푚 and [퐻 ] 푚 are nonsingular. Thus ∑ 푣 [퐵 ] 푚 is singular 푝 푝 푖=1 푖 푖∗∗ 푝 푛 if and only if ∑ 푤 [퐴 ] 푚 is singular, which implies 푆 is a semifield if and only 푗=1 푗 푗∗∗ 푝 [푚] ′ ′ if 푆[푚] is a semifield. And if 푆[푚] is a semifield, it is isotopic to 푆[푚] with isotopism

{[퐹 ]푝푚 ,[퐺]푝푚 ,[퐻]푝푚 }.

′ 푛 Corollary 7.4. Let 푆 and 푆 be isomorphic semifields of order 푝 . Then 푆[푚] is ′ a semifield if and only if 푆[푚] is a semifield. Further, if 푆[푚] is a semifield, it is ′ isomorphic to 푆[푚].

Now we address the obvious question of how to determine if [퐴]푝푚 is nonsingu-

lar. By deﬁnition, the only method to determine whether [퐴]푝푚 is nonsingular is to 푛 푛 check ∑ 푣 퐴 for all 푣 ∈ 픽 푚 . Consider the following, equivalent, method. Let 푖=1 푖 푖∗∗ 푝

푥 = (푥1, ..., 푥푛) be a vector of indeterminates, and deﬁne 푓 ∈ 픽푝[푥1, ..., 푥푛] by

푛

푓(푥1, ..., 푥푛) = det(푥퐴) = det (∑ 푥푖퐴푖∗∗) 푖=1

We will call 푓 the inﬂation polynomial of 퐴. Then [퐴]푝푚 is nonsingular if and only 푛 if 푓 contains no nontrivial zeroes in 픽푝푚 . Note that 푓 is a homogeneous polynomial 78 of degree 푛 in 푛 unknowns. While this does not make it easier to show that [퐴]푚 is nonsingular, the inﬂation polynomial can make it much easier to show that [퐴]푚 is singular for some given 푚.

7.2 Determining Valid Extension Fields

We will use the 16-element semifields to illustrate these results. As we mentioned in chapter 2, there are three isotopism classes for these semifields. Class 1 consists of 픽16, class 2 contains the semifields isotopic to system W, and class 3 contains the semifields isotopic to system V. Consider the following Knuth cubes, 퐴 and

퐵 which correspond to semiﬁelds in classes 2 and 3 respectively.

⎛ 1000 0100 00010 0001 ⎞ ⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 0100 1100 0001 0011 ⎟ ⎜ 0100 1100 0011 1110 ⎟ 퐴 = ⎜ ⎟ 퐵 = ⎜ ⎟ ⎜ 0010 0011 1010 1101 ⎟ ⎜ 0010 0001 0100 1100 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0001 0010 0101 1011 ⎠ ⎝ 0001 0011 1000 0111 ⎠

Let 푥 = (푎, 푏, 푐, 푑) be a vector of indeterminates. Then we get the following matrices, whose determinants will give us the inﬂation polynomials.

⎛ 푎 푏 푐 푑 ⎞ ⎜ ⎟ ⎜ 푏 푎 + 푏 푐 + 푑 푐 ⎟ 푥퐴 = ⎜ ⎟ ⎜ 푐 푑 푎 + 푐 푏 + 푑 ⎟ ⎜ ⎟ ⎝ 푐+푑 푐 푏+푑 푎+푏+푐+푑 ⎠ ⎛ 푎 푏 푐 푑 ⎞ ⎜ ⎟ ⎜ 푏 푎 + 푏 푑 푐 + 푑 ⎟ 푥퐵 = ⎜ ⎟ ⎜ 푑 푐 푎 + 푏 푏 ⎟ ⎜ ⎟ ⎝ 푏+푐 푏+푐+푑 푏+푑 푎+푑 ⎠

79 It turns out that these matrices have the same determinant, so we have the following

inﬂation polynomial for both 퐴 and 퐵:

푓(푎, 푏, 푐, 푑) = 푎4 + 푎3푑 + 푎2푏2 + 푎2푏푐 + 푎2푏푑 + 푎2푐2 + 푎2푐푑 + 푎2푑2 + 푎푏2푐 + 푎푏2푑

+푎푏푐2 + 푎푏푐푑 + 푎푏푑2 + 푎푐2푑 + 푎푐푑2 + 푎푑3 + 푏4 + 푏3푐 + 푏2푐2

+푏2푐푑 + 푏2푑2 + 푏푐3 + 푏푐2푑 + 푏푐푑2 + 푐4 + 푐2푑2 + 푑4

Without loss of generality, let 푆 denote the semifield defined by 퐴. We can find some 푚 for which 푆[푚] is not a semifield by setting all but one of the inputs equal to elements of 픽2:

푓(푥, 1, 1, 1) = 푥4 + 푥3 + 1

푓(푥, 1, 1, 0) = 푥4 + 푥2 + 1 = (푥2 + 푥 + 1)2

The ﬁrst polynomial is a primitive polynomial of 픽16, and thus has a root in 픽16,

implying that 푆[4] is not a semiﬁeld. The second polynomial is the square of the

primitive polynomial of 픽4 and thus 푆[2] is not a semiﬁeld. In fact, the second poly-

nomial also implies that 푆[4] is not a semiﬁeld, since 픽4 is isomorphic to a subﬁeld of

픽16. This can be generalized into the following lemma. 푛 Lemma 7.5. Let 푆 be a semiﬁeld of order 푝 . If 푆[푚] is not a semiﬁeld and 푚|푟, then

푆[푟] is not a semiﬁeld.

푛 푛 Proof. 푆 is not a semifield if for some 퐴 ∈ 퐾(푆) and 푣 ∈ 픽 푟 , ∑ 푣 퐴 is [푟] 푝 푖=1 푖 푖∗∗ 푛 푛 singular. Since 푆 is not a semifield, there exists 푤 ∈ 픽 푚 such that ∑ 푤 퐴 [푚] 푝 푖=1 푖 푖∗∗ 푛 is singular. Since 푚|푟, 픽푝푚 is isomorphic to a subfield of 픽푝푟 , and [푤]푝푟 ∈ 픽푝푟 . Thus

푆[푟] is not a semiﬁeld.

Corollary 7.6. If 푎 ∈ 픽푝푚 is a root of the characteristic polynomial of 퐿푧 for some

푧 ∈ 푆, then 푆[푚] is not a semiﬁeld.

Corollary 7.7. If 푎 ∈ 픽푝푚 is a root of the left minimal polynomial of any element of

푆, then 푆[푚] is not a semifield. 80 Returning to the 16-element semifields, we now know that 푆[2푘] is not a semifield for all 푘 ∈ ℤ+. By direct computation, 푓 was found to have no nontrivial zeroes in

4 4 4 픽8, 픽32, or 픽128. Thus 푆[3], 푆[5], and 푆[7] are all semiﬁelds. Furthermore, we know there exist at least three isotopism classes for semiﬁelds of order 212, 220, and 228. A key problem that remains is a general method for determining which extension

ﬁelds will not contain roots of the inﬂation polynomial. Based on Lemma 7.5, as well

as the properties of ﬁnite ﬁelds, the following conjecture may be the case.

Conjecture 7.8. Let 푆 be a semiﬁeld of order 푞푛, where 푞 is the order of the largest

subﬁeld of 푆 over which 푆 can be viewed as a vector space. Then 푆[푚] is a semiﬁeld if and only if 푚 and 푛 are relatively prime.

7.3 Results of Extension

The results in this chapter have focused on extending the prime subﬁeld of a

semifield, since every finite semifield can be viewed as a vector space over its prime

subfield. But, through inflation, we have shown that there exist semifields which can

be viewed as vector spaces over extensions of their prime subﬁeld. All of the results

presented in chapters 4 and 5 have been proven viewing 푆 as a vector space over its prime subﬁeld, but it is worth mentioning that similar results will hold if a diﬀerent

view of 푆 is taken. To illustrate this, we will let 푆 be the semiﬁeld of order 212 deﬁned

by [퐴]8 in the previous section. Then 푆 can either be viewed as a 12-dimensional

vector space over 픽2 or a 4-dimensional vector space over 픽8. First we consider the properties of left minimal polynomials of elements of 푆. If

푧 ∈ 푆 is left primitive, then, as we’ve deﬁned it, the left minimal polynomial of 푧 will be a primitive polynomial of degree 12 in 픽2[푥]. But, if we consider the characteristic polynomial of 푧[퐴]8, where 푧 is the vector form of 푧 over 픽8, then this polynomial

will be a primitive degree 4 polynomial in 픽8[푥]. 81 If 푝푛 = 푞푚, where 푞 is some power of 푝, we could have proven the theorems in

푚 푛 chapter 4 in terms of 픽푞 instead of 픽푝 , but it is more convenient to consider 푆 as a vector space over the prime subﬁeld. The results in chapter 5 could similarly be changed, looking at 퐶(푚, 푞) in place of 퐶(푛, 푝). If this is the case, then |퐶(푚, 푞)| will be less than |퐶(푛, 푝)|, but will likely still be too large to test.

82 Chapter 8

Determining Constructions of the Semiﬁelds of Order 16

In this chapter we will derive algebraic constructions for each of the 16-element semifields (up to isomorphism). As mentioned in chapter 6, the 16-element semifields were first enumerated by Kleinfeld in [10]. In his work, Kleinfeld essentially provided a Knuth cube for each isomorphism class of these semifields. We start by verifying these results, and in the process develop a database of Knuth cubes for the 16-element semifields. From there we determine which of the 16-element semifields correspond to known constructions in chapter 6. We then investigate the remaining semifields for which there is no known construction and develop a construction for each based on its Knuth cubes. We conclude with a full list of the 16-element semifields along with all of the information which can be determined for each based on the results from chapters 4 and 5.

8.1 Verifying Kleinfeld’s Results

We veriﬁed Kleinfeld’s results using a desktop PC and Wolfram Mathematica by determining all nonsingular Knuth cubes of dimension 4 over 픽2 using the following method.

As Kleinfeld proved in Theorem 3 of [10], given any 16 element semiﬁeld 푆,

2 2 there exists 푎 ∈ 푆 such that {1, 푎, 푎 , 푎(푎 )} forms a basis for 푆 over 픽2. The reverse

83 decimal matrix of a Knuth cube generated by this basis will then have the following

form: 1 2 4 8 ⎜⎛ ⎟⎞ ⎜ ⎟ ⎜ 2 4 8 푏1 ⎟ 퐴 = ⎜ ⎟ ⎜ ⎟ ⎜ 4 푏2 푏3 푏4 ⎟ ⎜ ⎟ ⎝ 8 푏5 푏6 푏7 ⎠

The unknowns, 푏푖, 1 ≤ 푖 ≤ 7, can each have values ranging from 1 to 15, subject to the fact that no value can correspond to a vector which is a linear combination of

the vectors corresponding to the other values in that row or column. For example, 푏1 must be odd, since any even value between 1 and 15 would be a linear combination

of the vectors corresponding to 2, 4, and 8. We had Mathematica construct reverse

decimal matrices of this form by trying all of the possible values in 퐴 starting with

the choice of 푏1, then 푏2, 푏5, 푏3, 푏4, 푏6, and ﬁnally 푏7. Once 퐴 was completed, the cubical array 퐴 corresponding to 퐴 was constructed and tested for nonsingularity. If

퐴 was nonsingular, it was added to a list, ℒ. Once all of the possible matrices had been constructed and their cubical arrays tested, we then needed to sort out the isomorphism and isotopism classes. For each

퐴 ∈ ℒ we construct a list ℒ퐴 of Knuth cubes generated by left cyclic bases generated by elements of the semiﬁeld 푆 deﬁned by 퐴. Each of these cubes will also be in ℒ,

so we remove them from ℒ to save time. We continue in this way until we have 24 lists of cubical arrays corresponding to the 24 isomorphism classes of the 16-element

semiﬁelds. Then, using the results from Theorem 3.18, we construct Knuth cubes for

all of the semiﬁelds isotopic to a Knuth cube from each isomorphism class. We then

sort these into the three isotopism classes. Finally, we use 퐶(4, 2) to generate all of the distinct Knuth cubes for each isomorphism class.

For each isomorphism class we assign a pair of numbers {푡, 푚}, where 푡 refers to

84 which isotopism class the semifield is included in, and 푚 refers to which isomorphism class it is within that isotopism class. In the database, {푡, 푚} is a list of up to 1344 reverse decimal matrices corresponding to the Knuth cubes for that isomorphism class. The choices of 푡 and 푚 are arbitrary and merely used as a convenient way of referencing the semifields as we investigate then. The following list identifies each isomorphism class in our database, DB, with the designation given by Kleinfeld, KD, with the exception of 픽16, which he did not include. System V is in {3, 71}, and system W is in {2, 5}.

DB KD DB KD DB KD DB KD

{1, 1} 픽16 {3, 1} V(5) {3, 8} V(7) {3, 74} V(4) {2, 1} T(24) {3, 2} V(9) {3, 9} V(18) {3, 75} V(8)

{2, 5} T(45) {3, 4} V(1) {3, 11} V(11) {3, 85} V(15)

{2, 6} T(50) {3, 5} V(3) {3, 70} V(10) {3, 88} V(2)

{2, 13} T(25) {3, 6} V(12) {3, 71} V(13) {3, 89} V(14)

{2, 15} T(35) {3, 7} V(17) {3, 73} V(16) {3, 90} V(6)

8.2 Eliminating Known Constructions

Using the results from chapter 6 we look at all of the semifields which can be generated using these known constructions. Rather than looking at each isomorphism class individually and determining whether it corresponds to a known construction, we instead simply construct the Knuth cubes for all possible constructions and find their locations in the database. First, note that many of the constructions do not allow for a 16-element case. There are clearly no Albert binary or Knuth binary semifields of order 16. There are also no proper Albert twisted fields of order 16, but this is not easily seen and would require a considerable amount of work to show.

Instead we simply note that this is a consequence of proposition 10.14 in [8]. 85 By building all of the possible Knuth cubes for the other constructions we

find that the following semifields are defined by the noted constructions, with QWN referring to the semifields which are quadratic over a weak nucleus.

Database Constructions

{2, 1} QWN

{2, 6} QWN

{2, 13} Petit, QWN

{2, 15} System W, Petit, QWN

{3, 70} QWN

{3, 71} System V, QWN This leaves 17 isomorphism classes for which there is no known construction. Giving a detailed description of the derivation of constructions for each of the 17 unknown semiﬁelds would be quite lengthy. For that reason we will limit the work presented here to a few noteworthy examples, and exclude the others that are derived in similar ways. The full results will be listed in section 8.6.

8.3 Determining Constructions Part 1: Quadratic Over 픽4 If the elements of a semifield of order 16 can be viewed as a two-dimensional left or right vector space over 픽4, then we will say such a semifield is left or right 퐷 quadratic over 픽4. Let 푆 be a semifield which is right quadratic over 픽4, and let 푆 퐷 be its dual. Note that 푆 would be left quadratic over 픽4, and, once a construction is found for 푆, it will be a simple matter to find 푆퐷 in the database and define its product through the use of an anti-isomorphism. For this reason we will focus only on semifields which are right vector spaces over 픽4 in this section and the next. 2 Suppose a 16-element semifield was right quadratic over 픽4 = {0, 1, 휔, 휔 = 86 1 + 휔} and had a basis {1, 휔, 휆, 휆휔} over 픽2, where (0 + 휆푏) ∗ (푐 + 휆0) = 휆(푏푐). Then a reverse decimal matrix corresponding to this basis would have the following form:

⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ 2 3 ⎟ ⎜ ⎟ ⎜ 4 8 ⎟ ⎜ ⎟ ⎝ 8 12 ⎠

The unknown semiﬁelds corresponding to {2, 5}, {3, 1}, {3, 2}, and {3, 85} have re-

verse decimal matrices of this form. We will focus on {2, 5}. There are 12 such

matrices in {2, 5}:

⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 3 9 14 ⎟ ⎜ 2 3 9 14 ⎟ ⎜ 2 3 11 13 ⎟ ⎜ 2 3 11 13 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 4 8 5 10 ⎟ ⎜ 4 8 5 10 ⎟ ⎜ 4 8 9 14 ⎟ ⎜ 4 8 10 15 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 8 12 15 5 ⎠ ⎝ 8 12 14 7 ⎠ ⎝ 8 12 7 9 ⎠ ⎝ 8 12 7 9 ⎠

⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 3 10 15 ⎟ ⎜ 2 3 10 15 ⎟ ⎜ 2 3 9 15 ⎟ ⎜ 2 3 9 15 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 4 8 13 6 ⎟ ⎜ 4 8 15 5 ⎟ ⎜ 4 8 5 11 ⎟ ⎜ 4 8 5 10 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 8 12 11 13 ⎠ ⎝ 8 12 9 14 ⎠ ⎝ 8 12 15 2 ⎠ ⎝ 8 12 14 3 ⎠

⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎛ 1 2 4 8 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 3 11 14 ⎟ ⎜ 2 3 11 14 ⎟ ⎜ 2 3 10 13 ⎟ ⎜ 2 3 10 13 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 4 8 3 9 ⎟ ⎜ 4 8 2 9 ⎟ ⎜ 4 8 3 14 ⎟ ⎜ 4 8 2 15 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 8 12 5 2 ⎠ ⎝ 8 12 5 3 ⎠ ⎝ 8 12 6 9 ⎠ ⎝ 8 12 7 9 ⎠

87 Let 퐴 be the cubical array corresponding to the ﬁrst of these matrices:

⎛ 1 2 4 8 ⎞ ⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 2 3 9 14 ⎟ ⎜ 0100 1100 1001 0111 ⎟ ⎜ ⎟ ∼ 퐴 = ⎜ ⎟ ⎜ 4 8 5 10 ⎟ ⎜ 0010 0001 1010 0101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 8 12 15 5 ⎠ ⎝ 0001 0011 1111 1010 ⎠

To determine the result of (푎 + 휆푏) ∗ (푐 + 휆푑) we use the properties of cubical arrays mentioned in chapter 3. Let 푢 = ⟨푎1, 푎2, 푏1, 푏2⟩ and 푣 = ⟨푐1, 푐2, 푑1, 푑2⟩ be the vector forms of (푎+휆푏) and (푐+휆푑) over 픽2. Also note that, for example, that 푎 = 푎1 +푎2휔. Let 푤 = 푣(푢퐴), which means 푤 is the3 vector form of (푎 + 휆푏) ∗ (푐 + 휆푑). We get the following result:

푤 = ⟨ 푎1푐1 + 푎2푐2 + 푏2푑2 + 푎2푑1 + 푏1푑1 + 푏2푑1 ,

푎2푐1 + 푎1푐2 + 푎2푐2 + 푏2푑1 + 푎2푑2 + 푏1푑2 ,

푏1푐1 + 푏2푐2 + 푎1푑1 + 푏1푑1 + 푏2푑1 + 푎2푑2 + 푏2푑2 ,

푏2푐1 + 푏1푐2 + 푏2푐2 + 푎2푑1 + 푏2푑1 + 푎1푑2 + 푎2푑2 + 푏1푑2 ⟩

88 Note that the ﬁrst two coordinates of 푤 are the coeﬃcients of 1 and 휔 respectively.

Thus we take 푤1 + 휔푤2 to ﬁnd the ﬁrst part of the product:

(1) 푎1푐1 + 푎2푐2 + 푏2푑2 + 푎2푑1 + 푏1푑1 + 푏2푑1

+휔(푎2푐1 + 푎1푐2 + 푎2푐2 + 푏2푑1 + 푎2푑2 + 푏1푑2)

(2) = 푎1푐1 + 푎2푐2 + 푏2푑2 + 푎2푑1 + 푏1푑1 + 푏2푑1 + 푎2푐1휔 + 푎1푐2휔 + 푎2푐2휔

+푏2푑1휔 + 푎2푑2휔 + 푏1푑2휔

(3) = (푎1푐1 + 푎2푐2 + 푎2푐1휔 + 푎1푐2휔 + 푎2푐2휔) + (푎2푑1 + 푎2푑2휔)

+(푏2푑2 + 푏1푑2휔 + 푏1푑1 + 푏2푑1 + 푏2푑1휔)

2 (4) = (푐1(푎1 + 푎2휔) + 푎1푐2휔 + 푎2푐2휔 ) + (푎2(푑1 + 푑2휔)) 2 +(푏2푑2 + 푏1(푑1 + 푑2휔) + 푏2푑1휔 )

2 (5) = (푐1(푎1 + 푎2휔) + (푐2휔)(푎1 + 푎2휔)) + (푎 + 푎 )푑 2 +(푏2푑2 + 푏1(푑1 + 푑2휔) + 푏2푑1휔 + 푏2푑2휔 + 푏2푑2휔)

2 (6) = ((푐1 + 푐2휔)(푎1 + 푎2휔)) + (푎 + 푎 )푑 2 +(푏1(푑1 + 푑2휔) + 푏2푑1 + 푏2푑1휔 + 푏2푑2휔 + 푏2푑2휔)

2 (7) = 푎푐 + (푎 + 푎 )푑 + (푏1(푑1 + 푑2휔) + (푏2휔)(푑1 + 푑2휔) + 푏2(푑1 + 푑2휔)) 2 (8) = 푎푐 + (푎 + 푎 )푑 + ((푏1 + 푏2휔)(푑1 + 푑2휔) + 푏2(푑1 + 푑2휔)) (9) = 푎푐 + (푎 + 푎2)푑 + 푏푑 + (푏 + 푏2)푑

(10) = 푎푐 + (푎 + 푎2)푑 + 푏2푑

89 2 On line 4 we have 푎2 isolated from 푎. The fact that 푎2 = 푎 + 푎 comes from the trace function acting on 푎. We now determine the second part of the product by looking at 푤3 + 휔푤4, which we will multiply on the left by 휆 when we are ﬁnished.

(1) 푏1푐1 + 푏2푐2 + 푎1푑1 + 푏1푑1 + 푏2푑1 + 푎2푑2 + 푏2푑2

+휔(푏2푐1 + 푏1푐2 + 푏2푐2 + 푎2푑1 + 푏2푑1 + 푎1푑2 + 푎2푑2 + 푏1푑2)

(2) = 푏1푐1 + 푏2푐2 + 푎1푑1 + 푏1푑1 + 푏2푑1 + 푎2푑2 + 푏2푑2

+푏2푐1휔 + 푏1푐2휔 + 푏2푐2휔 + 푎2푑1휔 + 푏2푑1휔 + 푎1푑2휔 + 푎2푑2휔 + 푏1푑2휔

(3) = (푏1푐1 + 푏2푐2 + 푏2푐1휔 + 푏1푐2휔 + 푏2푐2휔)

+(푎1푑1 + 푎2푑2 + 푎2푑1휔 + 푎1푑2휔 + 푎2푑2휔)

+(푏1푑1 + 푏2푑1 + 푏2푑2 + 푏2푑1휔 + 푏1푑2휔)

(4) = 푏푐 + 푎푑 + 푏2푑

This gives the ﬁnal result, where 푇 ∶ 픽4 ↦ 픽2 is the standard trace function 푇 (푥) = 푥 + 푥2:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푎)푑 + 푏2푑 + 휆(푏푐 + 푎푑 + 푏2푑)

90 Notice that this is very similar to the QWN construction, with the only diﬀerence

being the term 푇 (푎)푑. This is the result from looking at only the ﬁrst matrix, and it would be prudent to list products deﬁned by the other matrices:

(2) 푎푐 + 푇 (푎)푑 + 푏푑 + 휆(푏푐 + 푎푑 + 푏2푑)

(3) 푎푐 + 푇 (푎)푑휔2 + 푏2푑 + 휆(푏푐 + 푎푑 + 푏2푑휔)

(4) 푎푐 + 푇 (푎)푑휔2 + 푏푑휔 + 휆(푏푐 + 푎푑 + 푏2푑휔)

(5) 푎푐 + 푇 (푎)푑휔 + 푏2푑 + 휆(푏푐 + 푎푑 + 푏2푑휔2)

(6) 푎푐 + 푇 (푎)푑휔 + 푏푑휔2 + 휆(푏푐 + 푎푑 + 푏2푑휔2)

(7) 푎푐 + 푇 (푎)푑2 + 푏2푑2 + 휆(푏푐 + 푎푑 + 푏2푑 + 푇 (푏)푇 (푑))

(8) 푎푐 + 푇 (푎)푑2 + 푏푑 + 휆(푏푐 + 푎푑 + 푏2푑 + 푇 (푏)푇 (푑))

(9) 푎푐 + 푇 (푎)푑2휔2 + 푏푑휔2 + 휆(푏푐 + 푎푑 + 푇 (푏)푅(푑) + 푅(푏)푇 (푑)휔)

(10) 푎푐 + 푇 (푎)푑2휔2 + 푏2푑2휔 + 휆(푏푐 + 푎푑 + 푇 (푏)푅(푑) + 푅(푏)푇 (푑)휔)

(11) 푎푐 + 푇 (푎)푑2휔 + 푏2푑2휔2 + 휆(푏푐 + 푎푑 + 푏2푑휔 + 푅(푏)푅(푑)휔)

(12) 푎푐 + 푇 (푎)푑2휔 + 푏푑휔 + 휆(푏푐 + 푎푑 + 푏2푑휔 + 푅(푏)푅(푑)휔) where 푅(푎) = 푎1 + 푎2 + 푎1푎2. The initial choice is still one of the simplest representations (due to the lack of 푅 or products including 휔), so that is the one which is listed in section 8.6. For now we will also note that {3, 1}, {3, 2}, and {3, 85} yield formulas similar to (7) or (8) on the list.

In fact, {3, 1} will be useful to illustrate another approach to deriving constructions. First, note that the product formula for {3, 1} is

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푎)푇 (푑)휔2 + 푏2푑휔 + 휆(푏푐 + 푎2푑 + 푇 (푏)푇 (푑)휔2)

Let 퐴 be the cubical array which yielded this product. The transpose of 퐴 corresponds to a reverse decimal matrix located in {3, 74} meaning that these two semiﬁelds are

91 duals of each other. Thus we simply swap 푎 with 푐 and 푏 with 푑 in the product formula for {3, 1} to get the product for {3, 74}. This gives

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푏)푇 (푐)휔2 + 푏푑2휔 + 휆(푎푑 + 푏푐2 + 푇 (푏)푇 (푑)휔2)

This also means that the assumption (휆푏) ∗ 푐 = 휆(푏푐) will not be true for {3, 74}, and, instead 푏 ∗ (휆푐) = 휆(푏푐). Rather than describe the multiplication in this way, we will instead use the basis {1, 휔, 푋, 휔푋} since this behavior is similar to Petit’s construction. This changes the product in {3, 74} to

(푎 + 푏푋) ∗ (푐 + 푑푋) = 푎푐 + 푇 (푏)푇 (푐)휔2 + 푏푑2휔 + (푎푑 + 푏푐2 + 푇 (푏)푇 (푑)휔2)푋

This is how this semiﬁeld is described in section 8.6. And using this method, we can

ﬁnd {3, 73} and {3, 75}, which are the duals of {3, 85} and {3, 2} respectively.

8.4 Determining Constructions Part 2: Almost Quadratic Over 픽4

We will say a semifield of order 16 is almost right quadratic over 픽4 if it a basis of the form {1, 휔, 휆, 휆휔} where 휆 ∗ 휔 = 휆휔, but (휆휔) ∗ 휔 ≠ 휆(휔2). The semifields corresponding to {3, 7}, {3, 9}, {3, 11}, {3, 88}, and {3, 89} are all almost right quadratic over 픽4 are are the dual of such a semifield. We will only show the derivation for {3, 7} here. First, we choose the cubical array 퐴 corresponding to the following reverse decimal matrix:

⎛ 1 2 4 8 ⎞ ⎛ 1000 0100 0010 0001 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 2 3 14 4 ⎟ ⎜ 0100 1100 0111 0010 ⎟ ⎜ ⎟ ∼ 퐴 = ⎜ ⎟ ⎜ 4 8 6 3 ⎟ ⎜ 0010 0001 0110 1100 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 8 13 9 2 ⎠ ⎝ 0001 1011 1001 0100 ⎠

92 We let 푢 and 푣 be the vector forms of (푎 + 휆푏) and (푐 + 휆푑) over 픽2, and 푤 = 푣(푢퐴). This gives

푤 = ⟨ 푎1푐1 + 푎2푐2 + 푏2푐2 + 푏2푑1 + 푏1푑2 ,

푎2푐1 + 푎1푐2 + 푎2푐2 + 푎2푑1 + 푏1푑1 + 푏1푑2 + 푏2푑2 ,

푏1푐1 + 푏2푐2 + 푎1푑1 + 푎2푑1 + 푏1푑1 + 푎2푑2 ,

푏2푐1 + 푏1푐2 + 푏2푐2 + 푎2푑1 + 푏2푑1 + 푎1푑2 ⟩

Then the ﬁrst part of the product is 푤1 + 휔푤2:

(1) 푎1푐1 + 푎2푐2 + 푏2푐2 + 푏2푑1 + 푏1푑2

+휔(푎2푐1 + 푎1푐2 + 푎2푐2 + 푎2푑1 + 푏1푑1 + 푏1푑2 + 푏2푑2)

(2) = 푎1푐1 + 푎2푐2 + 푏2푐2 + 푏2푑1 + 푏1푑2

+(푎2푐1휔 + 푎1푐2휔 + 푎2푐2휔 + 푎2푑1휔 + 푏1푑1휔 + 푏1푑2휔 + 푏2푑2휔)

(3) = (푎1푐1 + 푎2푐2 + 푎2푐1휔 + 푎1푐2휔 + 푎2푐2휔) + (푏2푐2) + (푎2푑1휔)

+(푏2푑1 + 푏1푑2 + 푏1푑1휔 + 푏1푑2휔 + 푏2푑2휔)

2 (4) = 푎푐 + (푏2푐2) + (푎2푑1휔) + 푏 푑휔 (5) = 푎푐 + 푇 (푏)푇 (푐) + 푇 (푎)푅(푑)휔 + 푏2푑휔

93 where 푅(푎1 + 휔푎2) = 푎1. There is no standard function for 푅 like there was for 푇 ,

so we will leave it as is. Then we compute 푤3 + 휔푤4:

(1) 푏1푐1 + 푏2푐2 + 푎1푑1 + 푎2푑1 + 푏1푑1 + 푎2푑2

+휔(푏2푐1 + 푏1푐2 + 푏2푐2 + 푎2푑1 + 푏2푑1 + 푎1푑2)

(2) = 푏1푐1 + 푏2푐2 + 푎1푑1 + 푎2푑1 + 푏1푑1 + 푎2푑2

+(푏2푐1휔 + 푏1푐2휔 + 푏2푐2휔 + 푎2푑1휔 + 푏2푑1휔 + 푎1푑2휔)

(3) = (푏1푐1 + 푏2푐2 + 푏2푐1휔 + 푏1푐2휔 + 푏2푐2휔)

+(푎1푑1 + 푎2푑1 + 푎2푑2 + 푎2푑1휔 + 푎1푑2휔) + (푏1푑1 + 푏2푑1휔)

(4) = 푏푐 + 푎2푑 + 푅(푑)푏

And we get the following product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푏)푇 (푐) + 푇 (푎)푅(푑)휔 + 푏2푑휔 + 휆(푏푐 + 푎2푑 + 푅(푑)푏)

This method, and the use of duals, yields similar formulas for {3, 9}, {3, 11}, {3, 88},

and {3, 89}.

8.5 Determining Constructions Part 3: Not Quadratic Over 픽4 All of the semifields mentioned thus far have contained a subfield isomorphic to 픽4. The remaining unknown semifields, {3, 4}, {3, 5}, {3, 6}, {3, 8} and {3, 90} do

not posses this feature, and thus will not be quadratic or almost quadratic over 픽4. Rather than giving a product formula for such semiﬁelds, we will instead attempt

to find a basis of the form {1, 푎, 푏, 푐} for these semifields over 픽2 which will posses similar properties in each of these semifields. By investigation, all of these semifields 94 have such a basis where 푎2 = 푏, 푏2 = 푐, and 푐2 is some linear combination of 1, 푎, and 푏. In section 8.6, for each of these semifields we provide a multiplication table of the nontrivial basis elements in place of a product formula.

8.6 A Catalog Of The 16-Element Semiﬁelds

In this section we provide as much pertinent information as possible for each of the finite semifields of order 16. Each semifield, 푆, is given an entry in a similar format. First we list any names which have been given for 푆. Beneath the name, we provide a Knuth cube, 퐴, which was used to define the other results in the entry. To the right of 퐴 we list the database numbers for 푆, “DB”, the designation for 푆 given by

Kleinfeld, “KD”, the database numbers for the dual of 푆, “Dual DB”, the designation for the dual of 푆 given by Kleinfeld, “Dual KD”. Below the Knuth cube, we list which known constructions yield 푆, “Construction:”, we describe what form the elements may have, “Elements”, and below that provide a product formula or multiplication table. This is followed by the left, middle, and right nuclei, the center, and the left and right primitive elements of 푆 with respect to the given construction, denoted 푁푙,

푁푚, 푁푟, 푍, 푃푙 and 푃푟 respectively. This is followed by the lists of with respect to this construction, denoted 푃푙 and 푃푟 respectively. Finally, the automorphisms of 푆 are listed, denoted 휙푖 unless there is only the trivial automorphism. For each semiﬁeld which can be constructed using a known construction, we list the necessary information for that construction to be used. In the case where multiple variations of the same construction can be used, e.g. choices of 푓 qand 푔 in the QWN construction, we only provide one possible set of values. And in the case where multiple constructions yield the same semiﬁeld (up to isomorphism), the

95 construction used to deﬁne the product is presented last.

The results of chapters 4 and 5 were used to determine the primitive elements and automorphisms of each semiﬁeld. The center and nuclei of each semiﬁeld were determined by direct computation with the aid of a computer.

96 Description of the Semiﬁelds of Order 16

Name: 퐺퐹 (16), 픽16

⎛ 1000 0100 0010 0001 ⎞ DB: {1, 1} ⎜ ⎟ ⎜ 0100 1100 0001 0011 ⎟ Dual DB: {1, 1} 퐴 = ⎜ ⎟ ⎜ 0010 0001 0110 1101 ⎟ KD: None ⎜ ⎟ ⎝ 0001 0011 1101 1011 ⎠ Dual KD: None 2 Construction: The ﬁeld 픽4[휆], where 휆 is a root of 푥 + 푥 + 휔. 2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏푑휔 + 휆(푏푐 + 푎푑 + 푏푑)

푁푙 = 픽16

푁푚 = 픽16

푁푟 = 픽16

푍 = 픽16 2 2 2 푃푙 = {푎 + 휆|푎 ∈ 픽4} ∪ {1 + 휆휔, 휔 + 휆휔, 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {푎 + 휆|푎 ∈ 픽4} ∪ {1 + 휆휔, 휔 + 휆휔, 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

4 휙1(푎 + 휆푏) = 푎 + 휆푏 휙3(푎 + 휆푏) = (푎 + 휆푏) 2 8 휙2(푎 + 휆푏) = (푎 + 휆푏) 휙4(푎 + 휆푏) = (푎 + 휆푏)

97 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {2, 1} ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ Dual DB: {2, 1} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0101 ⎟ KD: T(24) ⎜ ⎟ ⎝ 0001 0011 1101 1011 ⎠ Dual KD: T(24) Construction: QWN: case 2, type 4, 푓 = 푔 = 1.

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔}

Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏푑)

2 푁푙 = {0, 1, 휔, 휔 } 2 푁푚 = {0, 1, 휆, 휆 } 2 푁푟 = {0, 1, 휔, 휔 } 푍 = {0, 1}

2 2 2 2 푃푙 = {1 + 휆휔, 휆 (휔), 휔 + 휆, 휆 (휔 )} 2 2 2 푃푟 = {1 + 휆휔, 휔 + 휆휔, 휔 + 휆, 휔 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 2 휙2(푎 + 휆푏) = 푎 + 휆(푏 )

98 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {2, 5} ⎜ ⎟ ⎜ 0100 1100 1001 0111 ⎟ Dual DB: {2, 5} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0101 ⎟ KD: T(45) ⎜ ⎟ ⎝ 0001 0101 1111 1010 ⎠ Dual KD: T(45) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푎)푑 + 푏2푑 + 휆(푏푐 + 푎푑 + 푏2푑),

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2. 2 푁푙 = {0, 1, 휔 + 휆, 휔 + 휆} 2 푁푚 = {0, 1, 휆, 휆 } 2 푁푟 = {0, 1, 휔, 휔 } 푍 = {0, 1}

2 2 2 푃푙 = {1 + 휆휔, 휔 + 휆휔, 1 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {1 + 휆휔, 휔 + 휆휔, 1 + 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

2 2 휙1(푎 + 휆푏) = 푎 + 휆푏 휙3(푎 + 휆푏) = 푎 + 휆(푏 ) 2 2 2 2 휙2(푎 + 휆푏) = 푎 + 휆 (푏) 휙4(푎 + 휆푏) = 푎 + 휆 (푏 )

99 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {2, 6} ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ Dual DB: {2, 13} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0101 ⎟ KD: T(50) ⎜ ⎟ ⎝ 0001 0011 1111 1010 ⎠ Dual KD: T(25) Construction: QWN: case 2, type 2, 푓 = 푔 = 1.

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏2푑)

2 푁푙 = {0, 1, 휆, 휆 } 2 푁푚 = {0, 1, 휔, 휔 } 2 푁푟 = {0, 1, 휔, 휔 } 푍 = {0, 1}

2 2 2 푃푙 = {1 + 휆휔, 휔 + 휆휔, 1 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 2 2 2 푃푟 = {휔 + 휆, 휔 + 휆, 휆 (휔), 휔 + 휆휔, 휔 + 휆(휔 ), 휆 (휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 2 휙2(푎 + 휆푏) = 푎 + 휆(푏 )

100 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {2, 13} ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ Dual DB: {2, 6} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0111 ⎟ KD: T(25) ⎜ ⎟ ⎝ 0001 0011 1101 1010 ⎠ Dual KD: T(50) Construction: Petit: 푀(푋) = 푋2 + 푋 + 1.

QWN: case 2, type 3, 푓 = 푔 = 1.

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏푑2)

2 푁푙 = {0, 1, 휔, 휔 } 2 푁푚 = {0, 1, 휔, 휔 } 2 푁푟 = {0, 1, 휆, 휆 } 푍 = {0, 1}

2 2 2 2 2 2 푃푙 = {휔 + 휆, 휔 + 휆, 휆 (휔), 휔 + 휆휔, 휔 + 휆(휔 ), 휆 (휔 )} 2 2 2 2 푃푟 = {1 + 휆휔, 휆 (휔), 1 + 휆(휔 ), 휆 (휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 2 휙2(푎 + 휆푏) = 푎 + 휆(푏 )

101 Description of the Semiﬁelds of Order 16

Name: System W

⎛ 1000 0100 0010 0001 ⎞ DB: {2, 15} ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ Dual DB: {2, 15} 퐴 = ⎜ ⎟ ⎜ 0010 0001 0100 1100 ⎟ KD: T(35) ⎜ ⎟ ⎝ 0001 0011 1000 0100 ⎠ Dual KD: T(35) Construction: Petit: 푀(푋) = 푋2 + 휔.

QWN: case 2, any type, 푓 = 휔, 푔 = 0.

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑휔 + 휆(푏푐 + 푎2푑)

2 푁푙 = {0, 1, 휔, 휔 } 2 푁푚 = {0, 1, 휔, 휔 } 2 푁푟 = {0, 1, 휔, 휔 } 푍 = {0, 1}

2 2 2 2 2 푃푙 = {휔 + 휆, 휔 + 휆, 휔 + 휆휔, 휔 + 휆휔, 휔 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 2 2 푃푟 = {휔 + 휆, 휔 + 휆, 휔 + 휆휔, 휔 + 휆휔, 휔 + 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏

휙2(푎 + 휆푏) = 푎 + 휆(푏휔) 2 휙3(푎 + 휆푏) = 푎 + 휆(푏휔 )

102 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 1} ⎜ ⎟ ⎜ 0100 1100 0011 1110 ⎟ Dual DB: {3, 74} 퐴 = ⎜ ⎟ ⎜ 0010 0001 0100 1100 ⎟ KD: V(5) ⎜ ⎟ ⎝ 0001 0011 1000 0111 ⎠ Dual KD: V(4) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푎)푇 (푑)휔2 + 푏2푑휔 + 휆(푏푐 + 푎2푑 + 푇 (푏)푇 (푑)휔2),

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {1 + 휆휔, 휔 + 휆휔, 1 + 휆(휔 ), 휔 + 휆(휔 ) 2 2 2 푃푟 = {휔 + 휆, 휔 + 휆, 휆휔, 1 + 휆휔, 휆(휔 ), 1 + 휆(휔 )} Automorphisms:

휙(푎 + 휆푏) = 푎 + 휆푏

103 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 2} ⎜ ⎟ ⎜ 0100 1100 0011 0110 ⎟ Dual DB: {3, 75} 퐴 = ⎜ ⎟ ⎜ 0010 0001 0100 1111 ⎟ KD: V(9) ⎜ ⎟ ⎝ 0001 0011 1000 0100 ⎠ Dual KD: V(8) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푎)푇 (푑)휔 + 푏2푑휔 + 휆(푏푐 + 푎2푑 + 푅(푏)푇 (푑)휔2), where 푇 (푎) = 푎 + 푎2, 푅(푎) = 푎 + 푇 (푎)휔; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2, 푅(푎) = 푎1.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {1 + 휆, 휔 + 휆, 1 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {휔 + 휆, 휔 + 휆, 휆휔, 1 + 휆휔, 휆(휔 ), 1 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 2 2 2 휙2(푎 + 휆푏) = 푎 + 푏 휔 + 휆(푏 휔 )

104 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 4} ⎜ ⎟ ⎜ 0100 0010 0101 1000 ⎟ Dual DB: {3, 90} 퐴 = ⎜ ⎟ ⎜ 0010 1111 0001 0100 ⎟ KD: V(1) ⎜ ⎟ ⎝ 0001 0101 1111 0110 ⎠ Dual KD: V(6) Construction:

Elements: (푎 + 푏푖 + 푐푗 + 푑푘), where 푎, 푏, 푐, 푑 ∈ 픽2 Product: Let 푧 = 1 + 푖 + 푗 + 푘. The products of 푖, 푗, and 푘 are:

∗ 푖 푗 푘

푖 푗 푖 + 푘 1

푗 푧 푘 푖

푘 푖 + 푘 푧 푖 + 푗

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

푃푙 = {1 + 푗, 푖 + 푗, 푗 + 푘, 푖 + 푗 + 푘}

푃푟 = {푖, 1 + 푖, 푘, 1 + 푘, 푗 + 푘, 1 + 푗 + 푘} Automorphisms:

휙(푎+푏푖+푐푗+푑푘) = 푎+푏푖+푐푗+푑푘

105 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 5} ⎜ ⎟ ⎜ 0100 0010 1001 1110 ⎟ Dual DB: {3, 8} 퐴 = ⎜ ⎟ ⎜ 0010 1111 0001 0100 ⎟ KD: V(3) ⎜ ⎟ ⎝ 0001 0011 0100 1100 ⎠ Dual KD: V(7) Construction:

Elements: (푎 + 푏푖 + 푐푗 + 푑푘), where 푎, 푏, 푐, 푑 ∈ 픽2 Product: Let 푧 = 1 + 푖 + 푗 + 푘. The products of 푖, 푗, and 푘 are:

∗ 푖 푗 푘

푖 푗 1 + 푘 1 + 푖 + 푗

푗 푧 푘 푖

푘 푗 + 푘 푖 1 + 푖

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

푃푙 = {1+푗,푖+푗,1+푗+푘,1+푖+푗+푘}

푃푟 = {푘, 푖 + 푘, 푗 + 푘, 1 + 푖 + 푗 + 푘} Automorphisms:

휙(푎+푏푖+푐푗+푑푘) = 푎+푏푖+푐푗+푑푘

106 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 6} ⎜ ⎟ ⎜ 0100 0010 1001 0101 ⎟ Dual DB: {3, 6} 퐴 = ⎜ ⎟ ⎜ 0010 0101 0001 1111 ⎟ KD: V(12) ⎜ ⎟ ⎝ 0001 1000 1101 0110 ⎠ Dual KD: V(12) Construction:

Elements: (푎 + 푏푖 + 푐푗 + 푑푘), where 푎, 푏, 푐, 푑 ∈ 픽2 Product: Let 푧 = 1 + 푖 + 푗 + 푘. The products of 푖, 푗, and 푘 are:

∗ 푖 푗 푘

푖 푗 1 + 푘 푖 + 푘

푗 푖 + 푘 푘 푧

푘 1 1 + 푖 + 푘 푖 + 푗

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

푃푙 = {푖, 1 + 푖 + 푗, 푖 + 푘, 푖 + 푗 + 푘}

푃푟 = {1+푗,1+푖+푗,1+푗+푘,푖+푗+푘} Automorphisms:

휙(푎+푏푖+푐푗+푑푘) = 푎+푏푖+푐푗+푑푘

107 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 7} ⎜ ⎟ ⎜ 0100 1100 0111 0010 ⎟ Dual DB: {3, 7} 퐴 = ⎜ ⎟ ⎜ 0010 0001 0110 1100 ⎟ KD: V(17) ⎜ ⎟ ⎝ 0001 1011 1001 0100 ⎠ Dual KD: V(17) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푏)푇 (푐) + 푇 (푎)푅(푑)휔 + 푏2푑휔 + 휆(푏푐 + 푎2푑 + 푅(푑)푏), where 푇 (푎) = 푎 + 푎2, 푅(푎) = 푎 + 푇 (푎)휔; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2, 푅(푎) = 푎1.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {1 + 휆, 휔 + 휆, 1 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {휆, 휔 + 휆, 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

휙(푎 + 휆푏) = 푎 + 휆푏

108 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 8} ⎜ ⎟ ⎜ 0100 0010 1111 0011 ⎟ Dual DB: {3, 5} 퐴 = ⎜ ⎟ ⎜ 0010 1001 0001 0100 ⎟ KD: V(7) ⎜ ⎟ ⎝ 0001 1110 0100 1100 ⎠ Dual KD: V(3) Construction:

Elements: (푎 + 푏푖 + 푐푗 + 푑푘), where 푎, 푏, 푐, 푑 ∈ 픽2 Product: Let 푧 = 1 + 푖 + 푗 + 푘. The products of 푖, 푗, and 푘 are:

∗ 푖 푗 푘

푖 푗 푧 푗 + 푘

푗 1 + 푘 푘 푖

푘 1 + 푖 + 푗 푖 1 + 푖

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

푃푙 = {푘, 푖 + 푘, 푗 + 푘, 1 + 푖 + 푗 + 푘}

푃푟 = {1+푗,푖+푗,1+푗+푘,1+푖+푗+푘} Automorphisms:

휙(푎+푏푖+푐푗+푑푘) = 푎+푏푖+푐푗+푑푘

109 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 9} ⎜ ⎟ ⎜ 0100 1100 1011 1110 ⎟ Dual DB: {3, 9} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 1101 ⎟ KD: V(18) ⎜ ⎟ ⎝ 0001 1011 1111 0110 ⎠ Dual KD: V(18) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푏)푇 (푐) + 푇 (푎)푑2 + 푏2푑2 + 휆(푏푐 + 푎2푑 + 푏2푑),

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {1 + 휆휔, 휔 + 휆휔, 1 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {휆휔, 휔 + 휆휔, 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 2 휙2(푎 + 휆푏) = 푎 + 휆(푏 )

110 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 11} ⎜ ⎟ ⎜ 0100 ⎟ Dual DB: {3, 89} 퐴 = ⎜ ⎟ ⎜ 0010 ⎟ KD: V(11) ⎜ ⎟ ⎝ 0001 ⎠ Dual KD: V(14) Construction:

2 Elements: (푎 + 푏푋) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 푏푋) ∗ (푐 + 푑푋) = 푎푐 + 푇 (푎)푇 (푑)휔2 + 푇 (푐)푏휔 + 푏푑2휔 + (푏푐 + 푎푑 + 푇 (푏)푇 (푑)휔)푋,

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1, 휔 + 휔푋, 휔2 + 휔푋}

2 2 2 푃푙 = {푋, 휔 + 푋, (휔 )푋, 휔 + (휔 )푋} 2 2 2 2 2 푃푟 = {휔푋, 1 + 휔푋, (휔 )푋, 1 + (휔 )푋, 휔 + (휔 푋), 휔 + (휔 )푋} Automorphisms:

휙1(푎 + 푏푋) = 푎 + 푏푋 휙3(푎 + 휆푏) = 2 휙2(푎 + 푏푋) = 푎 + 푏휔 + 푏푋 휙4(푎 + 휆푏) =

111 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 70} ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ Dual DB: {3, 70} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1001 0110 ⎟ KD: V(10) ⎜ ⎟ ⎝ 0001 0011 1110 1011 ⎠ Dual KD: V(10) Construction: QWN: case 2, type 1, 푓 = 1, 푔 = 휔.

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏2푑2휔)

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

휙1(푎 + 휆푏) = 푎 + 휆푏

휙2(푎 + 휆푏) = 푎 + 휆(푏휔) 2 휙2(푎 + 휆푏) = 푎 + 휆(푏휔 )

112 Description of the Semiﬁelds of Order 16

Name: System V

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 71} ⎜ ⎟ ⎜ 0100 1100 0011 0010 ⎟ Dual DB: {3, 71} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0111 ⎟ KD: V(13) ⎜ ⎟ ⎝ 0001 0011 1111 1001 ⎠ Dual KD: V(13) Construction: QWN: case 2, type 1, 푓 = 푔 = 1.

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product: (푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏2푑2)

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 2 2 2 푃푙 = {휔 + 휆, 휔 + 휆, 휆 (휔), 휔 + 휆휔, 휔 + 휆(휔 ), 휆 (휔 )} 2 2 2 2 2 2 푃푟 = {휔 + 휆, 휔 + 휆, 휆 (휔), 휔 + 휆휔, 휔 + 휆(휔 ), 휆 (휔 )} Automorphisms:

2 2 휙1(푎 + 휆푏) = 푎 + 휆푏 휙4(푎 + 휆푏) = 푎 + 휆(푏 ) 2 2 휙2(푎 + 휆푏) = 푎 + 휆(푏휔) 휙5(푎 + 휆푏) = 푎 + 휆(푏 휔) 2 2 2 2 휙3(푎 + 휆푏) = 푎 + 휆(푏휔 ) 휙6(푎 + 휆푏) = 푎 + 휆(푏 휔 )

113 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 73} ⎜ ⎟ ⎜ 0100 1100 0001 0011 ⎟ Dual DB: {3, 85} 퐴 = ⎜ ⎟ ⎜ 0010 1011 1010 1101 ⎟ KD: V(16) ⎜ ⎟ ⎝ 0001 1110 0111 1010 ⎠ Dual KD: V(15) Construction:

2 Elements: (푎 + 푏푋) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 푏푋) ∗ (푐 + 푑푋) = 푎푐 + 푇 (푐)푏2 + 푏푑2 + (푎푑 + 푏푐2 + 푏2푑)푋

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {휔 + 푋, 휔 + 푋, 휔푋, 1 + 휔푋, (휔 )푋, 1 + (휔 )푋} 2 2 2 2 2 푃푟 = {휔 + 푋, 휔 + 푋, 휔 + 휔푋, 휔 + 휔푋, 휔 + (휔 )푋, 휔 + (휔 )푋} Automorphisms:

휙1(푎 + 푏푋) = 푎 + 푏푋 2 2 휙2(푎 + 푏푋) = 푎 + (푏 )푋

114 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 74} ⎜ ⎟ ⎜ 0100 1100 0001 0011 ⎟ Dual DB: {3, 1} 퐴 = ⎜ ⎟ ⎜ 0010 0011 0100 1000 ⎟ KD: V(4) ⎜ ⎟ ⎝ 0001 1110 1100 0111 ⎠ Dual KD: V(5) Construction:

2 Elements: (푎 + 푏푋) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 푏푋) ∗ (푐 + 푑푋) = 푎푐 + 푇 (푏)푇 (푐)휔2 + 푏푑2휔 + (푎푑 + 푏푐2 + 푇 (푏)푇 (푑)휔2)푋

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {휔 + 푋, 휔 + 푋, 휔푋, 1 + 휔푋, (휔 )푋, 1 + (휔 )푋} 2 2 2 푃푟 = {1 + 휔푋, 휔 + 휔푋, 1 + (휔 )푋, 휔 + (휔 )푋} Automorphisms:

휙(푎 + 푏푋) = 푎 + 푏푋

115 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 75} ⎜ ⎟ ⎜ 0100 1100 0001 0011 ⎟ Dual DB: {3, 2} 퐴 = ⎜ ⎟ ⎜ 0010 0011 0100 1000 ⎟ KD: V(8) ⎜ ⎟ ⎝ 0001 0110 1111 0100 ⎠ Dual KD: V(9) Construction:

2 Elements: (푎 + 푏푋) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 푏푋) ∗ (푐 + 푑푋) = 푎푐 + 푇 (푏)푇 (푐)휔 + 푏푑2휔 + (푎푑 + 푏푐2 + 푇 (푏)푅(푑)휔2)푋 where 푇 (푎) = 푎 + 푎2, 푅(푎) = 푎 + 푇 (푎)휔; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2, 푅(푎) = 푎1.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {휔 + 푋, 휔 + 푋, 휔푋, 1 + 휔푋, (휔 )푋, 1 + (휔 )푋} 2 2 2 푃푟 = {1 + 푋, 휔 + 푋, 1 + (휔 )푋, 휔 + (휔 )푋} Automorphisms:

휙1(푎 + 푏푋) = 푎 + 푏푋 2 2 2 2 휙2(푎 + 푏푋) = 푎 + 푏 휔 + (푏 휔 )푋

116 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 85} ⎜ ⎟ ⎜ 0100 1100 1011 1110 ⎟ Dual DB: {3, 73} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1010 0111 ⎟ KD: V(15) ⎜ ⎟ ⎝ 0001 0011 1101 1010 ⎠ Dual KD: V(16) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푎)푑2 + 푏2푑 + 휆(푏푐 + 푎2푑 + 푏푑2),

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 2 2 푃푙 = {휔 + 휆, 휔 + 휆, 휔 + 휆휔, 휔 + 휆휔, 휔 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {휔 + 휆, 휔 + 휆, 휆휔, 1 + 휆휔, 휆(휔 ), 1 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 2 휙2(푎 + 휆푏) = 푎 + 휆(푏 )

117 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 88} ⎜ ⎟ ⎜ 0100 1100 1101 1011 ⎟ Dual DB: {3, 88} 퐴 = ⎜ ⎟ ⎜ 0010 0001 1100 1000 ⎟ KD: V(2) ⎜ ⎟ ⎝ 0001 1111 0111 1100 ⎠ Dual KD: V(2) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푏)푇 (푐)휔2 + 푇 (푎)푑휔2 + 푏2푑휔2 + 휆(푏푐 + 푎푑 + 푇 (푏)푅(푑)휔2), where 푇 (푎) = 푎 + 푎2, 푅(푎) = 푎 + 푇 (푎)휔; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2, 푅(푎) = 푎1.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

2 2 2 푃푙 = {휆휔, 1 + 휆휔, 휔 + 휆휔, 휔 + 휆휔, 휆(휔 ), 1 + 휆(휔 )} 2 2 2 2 푃푟 = {휆휔, 1 + 휆휔, 휔 + 휆휔, 휔 + 휆휔, 휔 + 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏

휙2(푎 + 휆푏) = 푎 + 푏휔 + 휆푏

118 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 89} ⎜ ⎟ ⎜ 0100 1100 0101 1111 ⎟ Dual DB: {3, 11} 퐴 = ⎜ ⎟ ⎜ 0010 0001 0100 1100 ⎟ KD: V(14) ⎜ ⎟ ⎝ 0001 1111 1000 0101 ⎠ Dual KD: V(11) Construction:

2 Elements: (푎 + 휆푏) where 푎, 푏 ∈ 픽4, and 픽4 = {0, 1, 휔, 휔 = 1 + 휔} Product:

(푎 + 휆푏) ∗ (푐 + 휆푑) = 푎푐 + 푇 (푏)푇 (푐)휔2 + 푇 (푎)푑휔 + 푏2푑휔 + 휆(푏푐 + 푎푑 + 푇 (푏)푇 (푑)휔),

2 where 푇 (푎) = 푎 + 푎 ; equivalently, if 푎 = 푎1 + 푎2휔, 푇 (푎) = 푎2.

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1, 휔 + 휆휔, 휔2 + 휆휔}

2 2 2 2 2 푃푙 = {휆휔, 1 + 휆휔, 휆(휔 ), 1 + 휆(휔 ), 휔 + 휆(휔 ), 휔 + 휆(휔 )} 2 2 2 푃푟 = {휆, 휔 + 휆, 휆(휔 ), 휔 + 휆(휔 )} Automorphisms:

휙1(푎 + 휆푏) = 푎 + 휆푏 2 휙2(푎 + 휆푏) = 푎 + 푏휔 + 휆푏

119 Description of the Semiﬁelds of Order 16

Name:

⎛ 1000 0100 0010 0001 ⎞ DB: {3, 90} ⎜ ⎟ ⎜ 0100 0010 1111 0101 ⎟ Dual DB: {3, 4} 퐴 = ⎜ ⎟ ⎜ 0010 0101 0001 1111 ⎟ KD: V(6) ⎜ ⎟ ⎝ 0001 1000 0100 0110 ⎠ Dual KD: V(1) Construction:

Elements: (푎 + 푏푖 + 푐푗 + 푑푘), where 푎, 푏, 푐, 푑 ∈ 픽2 Product: Let 푧 = 1 + 푖 + 푗 + 푘. The products of 푖, 푗, and 푘 are:

∗ 푖 푗 푘

푖 푗 푧 푖 + 푘

푗 푖 + 푘 푘 푧

푘 1 푖 푖 + 푗

푁푙 = {0, 1}

푁푚 = {0, 1}

푁푟 = {0, 1} 푍 = {0, 1}

푃푙 = {푖, 1 + 푖, 푘, 1 + 푘, 푗 + 푘, 1 + 푗 + 푘}

푃푟 = {1 + 푗, 푖 + 푗, 푗 + 푘, 푖 + 푗 + 푘} Automorphisms:

휙(푎+푏푖+푐푗+푑푘) = 푎+푏푖+푐푗+푑푘

120 References

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123 Biographical Statement

Kelly C. Aman was born in Moorhead, Minnesota, in 1983. He earned a B.A. in Mathematics from the University of Texas at Arlington in the spring of 2008. That summer he married his wife, Jacqueline, and that fall he began the Ph.D. program in the Department of MAthematics at UT Arlington. Kelly worked as a Graduate

Teaching Assistant from the fall of 2008 to the spring of 2012. From the fall of 2012 through the spring of 2014, he was awarded a GK-12 Fellowship, which gave him the unique experience of teaching aspects of his research to middle-school and high-school students.

The research which culminated in this dissertation began in 2011, and began to take focus in the spring of 2012. Kelly presented his ﬁrst results at Combinatorics

2012 in Perugia, Italy, and he went on to attend many more conferences throughout his ﬁnal two years in graduate school. Upon the completion of this dissertation, Kelly has had one paper published, along with three papers under review. The results in this dissertation are intended to be the groundwork for further research, and he plans to further investigate the topics in chapters 4 and 7.

His time as a GTA and GK-12 Fellow have inspired Kelly to become a teacher.

He believes that anyone can learn mathematics if they have the desire to learn, so part of a teacher’s duty is to share their passion for the subject with their students.

To that end, teaching will always be an important part of his life from this point forward.

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