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Discrete 208/209 (1999) 125–137 www.elsevier.com/locate/disc View metadata, citation and similar papers at core.ac.uk brought to you by CORE A survey of ÿnite semiÿelds provided by Elsevier - Publisher Connector M. Corderoa;∗, G.P. Weneb aDepartment of Mathematics, Texas Tech University, Lubbock, TX 79409, USA bDivision of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249, USA

Received 5 March 1997; revised 3 September 1997; accepted 13 April 1998

Abstract

The ÿrst part of this paper is a discussion of the known constructions that lead to inÿnite families of semiÿelds of ÿnite .The second part is a catalog of the known semiÿelds of ÿnite order. The bibliography includes some references that concern the subject but are not explicitly cited. c 1999 Elsevier Science B.V. All rights reserved.

MSC: primary 17A35; secondary 12K10; 20N5

Keywords: Semiÿelds; rings; Finite geometries

1. Introduction

A ÿnite semiÿeld is a ÿnite algebraic system containing at least two distinguished elements 0 and e. A semiÿeld S possesses two binary operations, and multi- plication, written in the usual notation and satisfying the following : (1) (S; +) is a with identity 0. (2) If a and b are elements of S and ab = 0 then a =0 or b=0. (3) If a, b and c are elements of S then a(b + c)=ab + ac and (a + b)c = ac + bc. (4) The e satisÿes the relationship ea = ae = a for all a in S. All semiÿelds will be ÿnite unless it is speciÿcally stated otherwise. It is seen that the addition is commutative and that there are unique solutions to the ax = b and xa = b for every nonzero a and every b in S. Furthermore, S has a vector structure over some prime ÿeld F = GF(p) and that S has pn elements where n is the of S over F.

∗ Corresponding author. E-mail address: [email protected] (M. Cordero)

0012-365X/99/$ - see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S0012-365X(99)00068-0 126 M. Cordero, G.P. Wene / 208/209 (1999) 125–137

The semiÿeld is due to Knuth [77]; semiÿelds are also called division rings and division (some authors do not insist that a division has a unity element). Any ÿnite ÿeld is an associative semiÿeld; the term proper semiÿeld will mean a semiÿeld in which the is not associative. Semiÿelds are also discussed in the texts of Dembowski [25], Hughes and Piper [52] and Stevenson [93]. The ÿrst part of this survey presents constructions leading to inÿnite families of semiÿelds. The second part catalogs individual semiÿelds by giving explicit construction and discussing the algebraic and geometric properties of each.

2. Inÿnite families

Any semiÿeld of order p or p2 must be a ÿnite ÿeld. Knuth [77] proved that a proper semiÿeld of least order has 16 elements by showing that the only semiÿeld of order 23 is the ÿnite ÿeld GF(23).

2.1. The early work of Dickson

The study of three-dimensional nonassociative division algebras began with Dickson [30] in 1905. He continued this study in [31,33,34] and gave a capsule summary in his monograph [35]. The ‘Dickson Commutative Division Algebras’ were introduced in the paper [32] and are of even dimension. The study of three-dimensional nonassociative division algebras was continued by Zelinsky [100] in his Master’s thesis. Menichetti [85,86] proved that any three- dimensional nonassociative division was a twisted ÿeld (see below) and derived formulae for counting the of these algebras [85]:  (q3 − q2 + q − 10)=3ifq≡1 (mod 3); V = (q3 − q2 + q − 6)=3 otherwise; where q is the number of elements in the ÿnite ÿeld. Kaplansky [73,74] applied Dickson’s construction over algebraically closed ÿelds and Harrison [46] studies commutative algebras three-dimensional over an algebraically closed ÿeld of zero. Wene [99] began a study of semiÿelds three- dimensional over ÿelds. The Dickson commutative semiÿelds are exhaustedly treated in the two papers by Burmester [12,13]. Let F be the ÿnite ÿeld GF(pn) where p is an odd prime and n¿1 and let f be any element of F that is not a . The semiÿeld S has the structure of a two-dimensional over F with elements 1 and .IfÂis an of F given by x = xpr ; 16r¡n; we can deÿne a multiplication in S by (a + b)(c + d)=(ac + f(bd)Â)+(ad + bc): The middle nucleus is the ÿeld F; the left and right nucleus is the subÿeld of F ÿxed by Â. M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137 127

These semiÿelds are discussed brie y on p. 195 in the text by Hughes an Piper [52].

2.2. Zemmer’s cyclic commutative semiÿelds

Zemmer in his 1952 paper [101] investigates semiÿelds with predetermined au- tomorphism group. He proved the following: Let n be a positive and F = GF(qk );q¿2;a Galois ÿeld with qk =2ns +1; for some positive integer s. Then there exists a commutative, proper semiÿeld A∗ of dimension 2n over F with the following properties: (i) A∗ has a cyclic basis relative to F. (ii) The of A∗ relative to F is the of order 2n. (iii) A∗ contains a unique associative of dimension n over F isomorphic to the ÿeld GF(qnk ).

2.3. The twisted ÿelds of Albert

The twisted ÿelds were introduced by Albert [2] in 1952 who continued to study these semiÿelds in a of papers [3–7]. Let F be the ÿnite ÿeld GF(pn) with n¿2. Let  and  be of F and let F and F be the subÿelds of F ÿxed by  and , respectively. The semiÿeld S has the structure of a vector space F over F ∩ F. Let e denote the element of F and let c be any element of F such that x y c 6= for any x; y ∈ F \{0}: x y

Deÿne linear transformations Ae and Be on F by

  xAe = x − cx and yBe = y − cy ; respectively:

  The product ∗ in S will be given by xAe ∗ yBe = xy − cx y . The vector space S becomes a semiÿeld with unit element e − c (see [6] for details). Kaplansky [74] conjectured: Any three-dimensional semiÿeld over a ÿnite ÿeld is associative or a twisted ÿeld. Menichetti proved this conjecture true in his papers [85,86].

2.4. Knuth’s binary semiÿelds

These commutative semiÿelds were constructed by Knuth [78] in 1965 from the ÿeld mn F = GF(2 ), where n is odd, n¿1. We follow Knuth’s construction. Let F0 be the m subÿeld GF(2 ). Considering F as a vector space over F0, let f be any nonzero linear from F to F0, i.e., f(a + b)=f(a)+f(b); 128 M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137 for all a; b ∈ F and all ;  ∈ F0. Deÿne a multiplication in F as follows: a ◦ b = ab +(f(a)b+f(b)a)2: Now we can deÿne a product ∗ in the vector space F by (1 ◦ a) ∗ (1 ◦ b)=a◦b: The vector space becomes a semiÿeld. Knuth then computes all the automorphisms and autotopisms of this semiÿeld, as well as the group of the corresponding .

2.5. Semiÿelds quadratic over a weak nucleus

A subÿeld K of a semiÿeld S is called a weak nucleus if (xy)z −x(yz)=0 whenever two of x; y and z are in K. The semiÿeld S will be a vector space over it’s weak nucleus but right and left multiplication may not be linear transformations over the weak nucleus. If the dimension of S as a vector space over it’s weak nucleus is two, we say that S is quadratic over a weak nucleus. Knuth [77] (1965) gave conditions to determine all isotopism classes of semiÿelds quadratic over a weak nucleus. The constructions begin with a ÿnite ÿeld F = GF(pn);n¿2. In all cases the semi- ÿeld S has the structure of a two-dimensional vector space over a ÿeld F with basis elements 1 and . The Case I semiÿelds are also called generalized Dickson semiÿelds. The construction requires that the characteristic not be two and that the automorphisms , ÿ and  are not all the identity automorphism. If the element f ∈ F is a nonsquare in F, the multiplication is given by (a + b)(c + d)=(ac + b dÿf)+(ad+bc): Dickson’s commutative semiÿelds form a special case and are the only commutative semiÿelds of this . Let F be the ÿnite ÿeld GF(pn) with automorphism  6= I and let f; g be any elements of F such that y+1 + gy − f 6= 0 for y ∈ F: The multiplication of a Case II semiÿeld is given by one of the following, where the product (a + b)(c + d) is given by

i. (ac + bd2 f)+(bc + ad + bdg) ii. (ac + bdf)+(bc + ad + bdg) iii. (ac + bd2 f)+(bc + ad + bdg) iv. (ac + bdf)+(bc + ad + bdg) and =−1. The semiÿelds constructed by Hughes and Kleinfeld [49] are a special case of this construction. The collineation group of the corresponding plane is computed in Hughes [50]. M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137 129

The concept of a weak nucleus is further exploited by Ganley [42] (1981) using the results in Cohen and Ganley [24]; new semiÿelds are discovered. We will describe these semiÿelds using the notation of Knuth [77]. The semiÿeld S has the structure of a two-dimensional vector space over F with basis elements 1 and  where F is the ÿnite ÿeld GF(3n). Multiplication is given by (a + b)(c + d)=(ac + g(b; d)) + (ad + bc + f(b; d)): The pair of functions g; f : F × F → F satisfy one of the following

Type f(b; d) g(b; d)

3 9 2 i. b dn1bd+n1n2bd 3 3 9 9 2 ii. b d n1b d + n1n2bd 3 3 9 2 9 iii. b d n1b d + n1n2bd iv. The dual of i: or iii:

2 In each case n1;n2 are nonsquares in F and = n1n2.

2.6. Sandler’s semiÿelds

These semiÿelds were constructed by Sandler [92] in 1962 from the ÿeld F = nm n GF(p ). We follow Sandler’s construction. Let F0 be the subÿeld GF(p ) and  pn the automorphism x 7→ x of F ÿxing the subÿeld F0. The semiÿeld is a vector space over F with basis 1;;2;:::;m−1, and with multiplication deÿned by xi = ixi : for all x ∈ F and i¡m and by

i j i j j ( a1)( a2)=a1 a2 for a1;a2 ∈F; where ij=i+j for i+j¡m and ij=(i+j) mod m for i+j¿m, where  ∈ F satisÿes no of degree ¡m over F0. Sandler goes on to compute the autotopism groups of these semiÿelds.

2.7. Kantor’s semiÿelds

A second of semiÿelds of characteristic two was discovered by Bill Kantor [72] in 1982. The construction parallels that of Knuth’s binary semiÿelds (see above). Let K = GF(q), where q =2r and let n¿2 with r(2n − 1)¿3. Let F be the ÿeld GF(q2n−1). Considering F as a vector space over K, let f be the trace map from F to K, i.e., 2Xn−2 f(a)= aqi ; i=0 130 M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137 for all a ∈ F. Deÿne a multiplication in F as follows: a ◦ b = a2b + f(ab)+f(b)a: For x ∈ F let x 7→ x be the inverse of the mapping x 7→ x ◦ 1. Now we can deÿne a product ∗ in the vector space F by a ∗ b = a2b + f(ab)+f(b)a: This semiÿeld has dimension r(2n − 1) over its GF(2). It remains an open question: When is a Kantor semiÿeld a Knuth binary semiÿeld?

2.8. The Boerner-Lantz semiÿelds

The construction is Theorem 3:2 of the 1986 paper of Boerner-Lantz [11]. Let q=pn with p¿3. Choose  ∈ GF(q) such that x2 −  is irreducible over GF(q) and 1 + 4 2 is a nonsquare. Identify GF(q )as{ 1+ 2a| 1; 2 ∈GF(q)}, where a 6∈GF(q)isa root of x2 −  = 0, and the semiÿeld S as { + ÿx | ; ÿ ∈ GF(q2)} where x 6∈GF(q2). Deÿne addition + on S to be the usual vector space addition. If multiplication ∗ is deÿned on S by

q q ( + ÿx) ∗ ( + x)= + ÿ( a − 1)+(  + ÿ )x;

2 where =1 +2a; 1;2 ∈GF(q), then S is a semiÿeld of dimension two over GF(q ).

2.9. Johnson’s derivable semiÿelds

Johnson in his 1988 paper [68] presents a construction for semiÿelds of order q4 be- ginning with a known semiÿeld of order q2. He shows that this construction generalizes the above construction of Boerner-Lantz [11].

2.10. The cyclic semiÿelds of Jha and Johnson

Jha and Johnson in their 1989 paper [57] use a geometric approach to construct a large collection of non-abelian semiÿelds, the cyclic semiÿelds, that coordinatize both the planes determined by the Hughes–Kleinfeld [49] and Sandler [92] semiÿelds. They also show that all semiÿelds of order q2, that are known to be non-central extensions of a subÿeld F = GF(q), are (up to and transpose) cyclic semiÿelds and ask: ‘Do there exist semiÿelds D which are non-central extensions of a subÿeld F such that (up to duality and=or transpose) D is not a cyclic semiÿeld?’

3. Explicit constructions

A complete catalog entry for a semiÿeld S should give a construction, describe the various nuclei, give the and isotopism classes, automorphism and M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137 131 autotopism groups and the collineation groups of the corresponding plane. Because of the many, many di erent isomorphism classes of semiÿelds, the geometers are content to work with isotopism classes since two semiÿelds coordinatize the same plane if and only if the are isotopic (see [77]). The translation planes are precisely those ane planes which can be coordina- tized by quasiÿelds; a quasiÿeld Q coordinatizing an ane plane A is a semiÿeld i the left nucleus of Q is isomorphic to the kernel of the corresponding ane plane.

The nuclei of a semiÿeld are the left nucleus, Nl = {x ∈ S | (xy)z − x(yz) = 0 for all y; z ∈ S}, the middle nucleus, Nm = {y ∈ S | (xy)z − x(yz) = 0 for all x; z ∈ S}, the right nucleus, Nr = {z ∈ S | (xy)z − x(yz) = 0 for all x; y ∈ S}, the nucleus, N = Nl ∩ Nm ∩ Nr and the , Z = {n ∈ N | nx = xn for all x ∈ S}. Di erent semiÿelds coordinatizing isomorphic planes have isomorphic nuclei; how- ever, semiÿelds with isomorphic nuclei do not necessarily coordinatize isomorphic planes. Jha and Johnson [55] have shown that the center of a ÿnite semiÿeld is a geometric invariant. Many of the geometric properties of the corresponding plane, the semiÿeld plane, have been investigated. The of the corresponding matrices have been computed in several cases and the number of ovals has been computed in certain cases (see [25]). It is well known that the rank of the incidence of an ane plane coordinatized  n n p+1 by the ÿeld GF(p )is 2 and the corresponding rank of the will  n p+1 n be 2 + 1. The rank of the plane coordinatized by a semiÿeld of p elements depends on exactly which semiÿeld is used.

3.1. Order 16

Kleinfeld [75] computed all 16-element semiÿelds; there are 24 such semiÿelds. One is the ÿeld GF(16); the 23 proper semiÿelds form two isotopism classes: ÿve have nucleus (isomorphic to) GF(4) and 18 have nucleus (isomorphic to) GF(2). The translation ovals have been computed by Cherowitzo and checked by Jha and Wene. One plane is that coordinatized by the ÿeld GF(16). This plane has an of rank 81. There are 30 translation ovals. All 18 semiÿelds with nucleus GF(2) are isotopic to system V in [77]. The incidence matrix has rank 97 and there are no translation ovals. Let F be the ÿeld GF(4) with elements 0; 1;! and !2 = ! + 1. The elements of V are of the form a + b where a; b ∈ F. Addition and multiplication in V are deÿned in terms of the addition and multiplication of F.

(a + b)+(c+d)=(a+c)+(b+d);

(a+b)(c + d)=(ac + b2d)+(bc + a2d + b2d2): 132 M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137

The ÿeld F is a weak nucleus. The system V possesses six automorphisms ij given by

i i j (a + b)ij = a + ! b for i =0;1;2; and j =1;2: No other semiÿeld of order 16 has as many automorphisms: the ÿeld GF(16) has only four automorphisms. V is also antiisomorphic to itself under the mapping (a + b) = a + b2. All 16-element semiÿelds with nucleus GF(4) are isotopic to system W in Knuth [77]. The incidence matrix has rank 97 and there are 18 translation ovals. The con- struction paralleled that of System V , the di erence being that multiplication is now deÿned by

(a + b)(c + d)=(ac + !b2d)+(bc + a2d):

It is an easy computation to see that Nl = Nm = Nr are isomorphic to F. This system 2 has three automorphisms i given by (a + b)i = a + ib for i =1;!;! :

3.2. Order 27

The ÿve proper semiÿelds of order 33 belong to one isotopism class, see [31] for explicit constructions of all ÿve. Each is an algebra with a basis {1;i;j} over the ÿeld GF(3). The commutative semiÿeld has multiplication determined by (1) 1 is the unit element. (2) i · i = j, (3) i · j = j · i =1+i, and (4) j · j =2+i+j. See also [97].

3.3. Order 32

Walker [95] computed one representative of each of the six isotopism classes of 32 element semiÿelds. One is the ÿeld GF(32); the others are proper semiÿelds. Walker also computed all 24 of the commutative semiÿelds of this order; see also [78]. Wene and Jha (unpublished) computed the number of translation ovals in each of the six isotopism classes.

3.4. Order 64

The class of semiÿelds of order 64 includes the binary semiÿelds and semiÿelds quadratic over a weak nucleus that were discussed earlier. Huang and Johnson [48] (1990) compute all eight semiÿelds corresponding to the translation planes of order 64 and kernel containing GF(8) that admit a collineation group of order 2 · 64 in the linear translation complement. M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137 133

3.5. Order 81

Knuth’s construction in [77] can be used to produce semiÿelds quadratic over a weak nucleus isomorphic to GF(9): Cordero [15] explicitly constructs all p-primitive semiÿelds of this order. Denote the elements of F = GF(3) by {0; 1; 2}. Let m be a root of the x2 +1=0 over F. Then GF(9) ≈ F(m). The element m + 1 is a nonsquare in F(m). Let S = {a + b | a; b ∈ F(m)}. The addition of S is the vector space addition; the multiplication of a Dickson commutative semiÿeld will be (a + b)(c + d)=ac +(m+ 1)(bd)3 + (ad + bc):

The middle nucleus is isomorphic to GF(9); Nl = Nr ≈ GF(3). Burmester [12] gives a commutative semiÿeld of order 81 that is not a Dickson commutative semiÿeld. Let S be the same as in the Dickson commutative semiÿeld; only the multiplication will be given by (a + b)(c + d)=ac +(bd)3 +(2m+2)bd + (ad + bc +2(bd)3 +2bd):

The middle nucleus is GF(9); Nl = Nr ≈ GF(3). Boerner-Lantz [11] uses her method to construct a noncommutative semiÿeld of this order. Denote the elements of F = GF(3) by {0; 1; 2}. Let f be a root of the equation x2 + x +2=0 over F. Then GF(9) ≈ F(f). Let S have the additive structure as in the Dickson commutative semiÿeld; only the multiplication will be given by

3 3 (a + b)(c + d)=ac + b(d f − d1)+(ad + bc ) where d = d1 + d2f, d1;d2 ∈F. In this case, we have Nl ≈ GF(9), Nm ≈ GF(3) and Nr ≈ GF(3):

3.6. Order 125

There are 34 semiÿelds with 125 elements; by Menichetti’s formula [86] there are 33 proper semiÿelds. Long [83] found 32 noncommutative and two commutative semi- ÿelds.

3.7. Order p3

These semiÿelds form a special case of the three-dimensional Dickson algebras dis- cussed in Section 2.1.

3.8. Order p4 p-primitive planes

These are the semiÿeld planes of order p4 and kernel GF(p2) with the property that they admit a p-primitive Baer collineation. In [15] Cordero has shown the following results: 134 M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125–137

Theorem 3.1. Let (f0;f1) be a p-primitive plane and let Nl;Nm;Nr be its left; middle and right nuclei respectively. Then exactly one of the following holds: 2 (1) Nl = Nm = Nr ≈ GF(p ); or (2) Nm = Nr ≈ GF(p). Moreover we have

(1) holds ⇔ f0 =0 and (2) holds ⇔ f0 6=0.

p+1 2 Theorem 3.2. For any odd prime p; there are [ 2 ] nonisomorphic p-primitive planes of order p4.

The autotopism groups are determined for the general case. The corresponding planes are classiÿed and the number of nonisomorphic semiÿelds are determined for the primes p =3;5;7 and 11.

4. For further reading

The following references are also of interest to the reader: [1,8–10,14,16–23, 26–29,36–41,43–45,47,51,53,54,56,58–67,69–71,76,79–82,84,87–91,94,96,98].

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