On Finding Totally Real Quintic Number Fields of Minimal Signature Group Rank

ON FINDING TOTALLY REAL QUINTIC NUMBER FIELDS OF MINIMAL SIGNATURE GROUP RANK

A Thesis Presented

by Jason B. Hill to The Faculty of the Graduate College of The University of Vermont

In Partial Fulﬁllment of the Requirements for the Degree of Master of Science Specializing in Mathematics

April 6, 2006 Accepted by the Faculty of the Graduate College, The University of Vermont, in partial fullﬁllment of the requirements for the degree of Master of Science, specializing in Mathe- matics.

Thesis Examination Committee:

Advisor David S. Dummit, Ph.D.

Jonathan W. Sands, Ph.D.

Chairperson Christian Skalka, Ph.D.

Vice President for Frances E. Carr, Ph.D. Research and Dean of the Graduate College

Date: April 6, 2006 Abstract

An algorithm is created to find the totally real quintic number field(s) of smallest field discriminant satisfying a given property. This algorithm is efficient and improves upon current number field table building techniques in the sense that only computations on totally real fields are ever considered. The algorithm is then used to locate totally real quintic number fields of all signature group ranks, including the first known instance of a

ﬁeld with a totally positive system of fundamental units. Acknowledgements

I would like to thank David S. Dummit, my advisor, for his assistance and guidance. I also wish to thank my thesis defense committee members Christian Skalka and Jonathan

W. Sands.

Special thanks should be given to Michael E. Pohst for several conversations that helped my research greatly. Also, I am appreciative of the eﬀorts of Larry Kost in securing computing resources on campus. In addition, I thank my father Jeﬀrey Hill and the mathematics graduate students at The University of Vermont for their continued support.

ii Table of Contents

Acknowledgements ...... ii

Chapter

1. Background and Motivation ...... 1

2. Introduction ...... 4

3. Hunter’s Theorem ...... 10

4. Generating Totally Real Quintic Number Fields ...... 16

Placing a Lower Bound on a2 ...... 18

Placing an Upper Bound on a2 ...... 19

Bounding a3 with Polynomial Discriminants ...... 20

Bounding a4 with Minima and Maxima ...... 21

Bounding a5 with Minima and Maxima ...... 23

An Algorithm for Generating Totally Real Quintic Number Fields . . . . 24

5. Tables of Quintic Number Fields of Small Discriminant ...... 28

6. Finding Fields of Smallest Discriminant ...... 31

7. Examples Fields of Smallest Discriminant ...... 35

8. Totally Real Quintic Number Fields of Minimal Signature Group Rank . 37

The Smallest Discriminant Rank 5 Field...... 37

The Smallest Discriminant Rank 4 Field...... 37

The Smallest Discriminant Rank 3 Field...... 38

An Example of a Rank 2 Field...... 38

An Example of a Rank 1 Field...... 40

9. Conclusion ...... 44

References ...... 45 Background and Motivation 1

1 Background and Motivation

Suppose K is a Galois extension of a ﬁeld k with an abelian Galois group G. The Stark

Conjectures assert that there are connections between the (algebraically deﬁned) group of

units of K and the (analytically deﬁned) Artin L-series of k deﬁned by K. More precisely,

suppose that S is a set of places in k containing the Archimedean primes and the ﬁnite

primes that are ramiﬁed in K/k, and suppose further (for simplicity) that S contains at

least 3 places. For any character χ of G, let LS(s, χ) denote the (imprimitive) Artin L-series defined by −1 Y χ(σp) L (s, χ) = 1 − S Nps (p,S)=1 where p ranges over all (finite) prime ideals not contained in the set S and χ is viewed as an ideal character by class field theory. The usual ‘rank one abelian Stark Conjecture’ (first formulated in the 1970’s) says that if S contains a place v that splits completely from k to

K, then there should exist an element in K that evaluates the derivative at s = 0 of these

L-series: 1 X L0(0, χ) = − χ(σ) ln |σ| . e w σ∈G where e denotes the number of roots of unity in K. In short, there should be an “L-function evaluator”. In the case where v is an Archimedean prime, the evaluator should in fact be a unit, the so-called “Stark unit” for K. In addition to providing a closed form expression for the value of these L-series, to account for ‘extra factors of 2,’ Stark further hypothesized an “abelian condition” that K(1/e) should be an abelian extension of k (a priori, it would be only metacyclic).

The Stark Conjectures are extremely important fundamental questions in algebraic number theory. Background and Motivation 2

In a conversation with Dummit in October, 1994, Stark suggested that few computations

had been done on Stark’s Conjecture testing its “functoriality” (very much not the word

Stark would have used), i.e., the compatibility of various Stark units when their ﬁelds are

subﬁelds of a common ﬁeld. As early as 1995, while involved with Sands and Tangedal

in computations relating to Stark’s Conjecture in the case where k is a totally real cubic

ﬁeld, Dummit observed that it was possible to construct examples similar to the classical

‘rank one’ situation (namely, in which all the characters were of rank 1, i.e., had a zero

at s = 0 or order at least 1), but where there was no place v that split completely in the extension K/k. Over the course of several years, Dummit formulated (but did not publish, although some lectures on the topic were given in 1997-2000) a “robust Stark

Conjecture”, mentioning this work to Stark in 1998. The feature of a single unit serving as an “L-function evaluator” is lost in Dummit’s version. In preparing a talk delivered at the

Mathematical Sciences Research Institute in Berkeley in December 2001, Stark had the idea that this evaluator feature could be recovered in an ‘extended’ version of the Conjecture, still forgoing the assumption of a totally split prime. In discussing this extension with

Dummit late in 2003, unpublished computations by Dummit provided a counterexample to the original formulation. A revised ‘extended Conjecture’ was then formulated by Stark, and this version was considered by Stark’s student Erickson in his 2004 Ph.D. dissertation.

There are diﬀerences between the ‘robust Stark Conjecture’ and the ‘extended Stark

Conjecture’, and the precise relation between them has not been resolved. For example, the

“L-function evaluator” is present in the extended version and absent in the robust version, but the “abelian condition” is present in the robust version and absent (by examples of

Erickson) in the extended version. To test both conjectures it is necessary to construct

fields k having abelian extensions K with the property that every character χ of the Galois group G has rank at least one, and such that the difference between the class group and the strict class group of k is as large as possible. The first examples would occur for a totally Background and Motivation 3

real quintic base ﬁeld k.

The diﬀerence between the class group and the strict class group of k is determined

by the size of the signature group of the units in k, with the largest diﬀerence occurring

when k has a basis of fundamental units which are totally positive (i.e., rank of signature

group 1). No examples of such a totally real quintic are provided in the existing literature.

In fact, signature groups of rank 2 are not available. Two examples of rank 3 signature

groups for totally real quintics provide evidence for both conjectures, but are not suﬃcient

to distinguish between them or to test them beyond the evidence already known. Also, the

application to Stark’s Conjecture involves the computation of values of derivatives of L-

series to extremely high precision, so it is important to construct examples with a minimal

possible ﬁeld discriminant, since this is the primary limiting factor in the computational

time required.

In addition to the interest in ﬁnding totally real quintic number ﬁelds with a totally

positive system of fundamental units for application to considerations of Stark’s Conjecture,

the question of the distribution of unit group signature ranks for number ﬁelds of given

degree and signature type (e.g., totally real) is of intrinsic interest. The question of “how

many” number ﬁelds there are with a given signature structure for their unit groups is

analogous to the question of how many number ﬁelds there are with a given class group,

the ‘answer’ to which is provided by the so-called “Cohen-Lenstra” heuristics. No such

heuristics appear to have been formulated in the case of unit groups. The purpose of this

thesis is to provide examples of totall real quintic ﬁelds with small unit signature groups

and to provide numerical data on which a heuristic might be formulated. Introduction 4

2 Introduction

We ﬁrst set our notation. Let K = Q(α) be a number ﬁeld of degree n over Q generated by

α. Denote the discriminant and ring of integers of K by d(K) and OK , respectively. The

unit group of K is by deﬁnition the group of units in the ring OK and is given by

U = { ∈ O |N () = ±1}. K K K/Q

The torsion elements in UK are given by the roots of unity in K, a ﬁnite cyclic subgroup µ(K) generated by a root of unity ζ. Let [r, s] denote the signature of the ﬁeld K, given by the number r of real embeddings ρ : K → R and the number s of pairs σ, σ : K → C of complex conjugate embeddings. Dirichlet’s unit theorem states that the group of units UK is isomorphic to the direct product of µ(K) and a free abelian group of rank t = r + s − 1.

Then there exist units 1, 2, ..., t in UK called fundamental units such that any other unit in UK can be written uniquely as a product

v0 v1 v2 vt = ζ 1 2 ··· t

with integers vi where 0 ≤ v0 < |µ(K)|.

In the case of totally real quintic number ﬁelds we have d(K) > 0 and µ(K) is simply

{±1} and is generated by ζ = −1. Since in this case the signature of K is given as [5, 0], ∼ the unit group UK will be a direct product of µ(K) = Z2 with a rank 4 free abelian group

generated by algebraic integers 1, 2, 3 and 4. Consider the 5 × 5 matrix M1 formed by Introduction 5

taking the (5 real) embeddings of ζ and each fundemental unit i, i.e.,

   −1 sign(1) sign(2) sign(3) sign(4)         −1 sign(σ1(1)) sign(σ1(2)) sign(σ1(3)) sign(σ1(4))      M =   1  −1 sign(σ2(1)) sign(σ2(2)) sign(σ2(3)) sign(σ2(4))         −1 sign(σ ( )) sign(σ ( )) sign(σ ( )) sign(σ ( ))   3 1 3 2 3 3 3 4      −1 sign(σ4(1)) sign(σ4(2)) sign(σ4(3)) sign(σ4(4))    −1 ±1 ±1 ±1 ±1         −1 ±1 ±1 ±1 ±1        =  −1 ±1 ±1 ±1 ±1  .        −1 ±1 ±1 ±1 ±1        −1 ±1 ±1 ±1 ±1

Then M1 is a matrix over the multiplicative group Z2. We deﬁne the matrix M2 over

the additive group Z/2Z as follows. Let

  0 if M1(i, j) = 1 M2(i, j) = .  1 if M1(i, j) = −1

The columns of M2 then span a vector space over the ﬁnite ﬁeld F2. The rank of the

signature group of K is then deﬁned as the (row) rank of M2. Introduction 6

As an example, consider the totally real quintic number ﬁeld K of discriminant 638,597

generated by a root α of the polynomial

p(X) = X5 − 2X4 − 6X3 + 8X2 + 8X + 1.

A system of fundemental units for K is given by

1 = α,

2 2 = α − 2α,

2 3 = α − 2α − 2

4 3 2 and 4 = α − 2α − 6α + 9α + 7.

Calculating M1 and M2 gives

     −1 −1 1 1 −1   1 1 0 0 1               −1 −1 1 −1 1   1 1 0 1 0              M1 =  −1 −1 1 −1 1  and M2 =  1 1 0 1 0  .              −1 1 1 −1 1   1 0 0 1 0              −1 1 1 −1 1 1 0 0 1 0

For example, the roots σi(α) of p(X) have approximate numerical values

−2.0201, −0.5931, −0.1498, 2.0638 and 2.6992

2 so the ﬁve real embeddings of 2 = α − 2α are given approximately as

8.121, 1.537, 0.322, 0.131 and 1.887, Introduction 7

whose signs give the second and third columns of M1, respectively.

Therefore, noting that M2 has (row) rank 3, K has signature group rank 3.

At present, available literature gives the existence of totally real quintic number ﬁelds

of signature group ranks 3, 4 and 5, corresponding to vector spaces of size 8, 16 and 32,

respectively, over F2. However, only two examples of rank 3 ﬁelds are provided.

Our main task here is to ﬁnd totally real quintic number ﬁelds of small signature group

rank. Ultimately, we desire a rank 1 ﬁeld, that is, a totally real quintic number ﬁeld with

a totally positive system of fundamental units. In addition, we desire that the ﬁelds under

consideration be of the smallest possible discriminant. Our primary goal can thus be stated

explicitly as the following: Given a signature group rank, ﬁnd, or at least develop a method

for ﬁnding, the totally real quintic number ﬁeld(s) of that signature group rank having the

smallest discriminant, assuming one exists.

In order to verify that some number ﬁeld K of signature [r, s] is in fact the ﬁeld of

smallest absolute discriminant satisfying some property, we require the ability to test all

number ﬁelds (up to isomorphism) of the same signature and smaller absolute discriminant

to determine that the property fails to hold. At present, this is accomplished by using a

so-called “table building” approach. Due to a theorem of J. Hunter, it is possible to bound

the coeﬃcients of the characteristic polynomial of some non-rational algebraic integer inside

every number ﬁeld of a given degree satisfying certain discriminant bounds. It is therefore

possible, although typically cumbersome, to create a table of number ﬁelds by considering

the ﬁelds generated by the roots of all polynomials satisfying these bounds. Introduction 8

Thus, if we are provided with one example of a totally real quintic number ﬁeld K of

a given signature group rank, we may use d(K) as an upper discriminant bound for table

building. In this case, we can construct a table of all totally real quintic number ﬁelds

having discriminant less than d(K) and checking the signature group rank of each ﬁeld in

the table gives the desired result.

However, when no instance of a ﬁeld is known satisfying the property in question (e.g.,

with signature group rank 1), no initial discriminant bound can be given. Although a ﬁeld

satisfying the property is not provided as a starting point, the approach of table building may

still be useful. One possibility is that exhaustive tables of totally real quintic number ﬁelds

satisfying increasing discriminant bounds may be recursively calculated until an instance of

the desired property is found in some ﬁeld K. Then, we use d(K) as an upper discriminant

bound and proceed as described previously.

The process of number ﬁeld table building typically begins by a desire to construct all

ﬁelds of a certain degree satisfying some absolute discriminant bound. As a result, bounds

for table building algorithms are generally calculated before the algorithms are executed.

This makes enlarging an existing table of number ﬁelds to account for some larger dis-

criminant bound challenging (at least if we want to avoid repeating calculations), and thus

the idea discussed in the previous paragraph for recursively generating our tables becomes

complicated. Also, ﬁelds of all signatures will be generated, meaning that calculation time

will be wasted on ﬁelds that are not pertinent to our speciﬁc search. Thus, if we wish to

concentrate on totally real quintic number ﬁelds and we cannot give a discriminant bound

before our search begins, we ﬁnd that available table building techniques must be modiﬁed.

We will discuss a modiﬁcation of standard table building techniques to make our current

search more eﬃcient. We ﬁnd that it is possible to modify table building algorithms based Introduction 9

on Hunter’s theorem so that no initial discriminant bound need be given in the case of

totally real quintic number ﬁelds. In addition, we can restrict our table building algorithm

in such a way that only calculations over totally real ﬁelds are ever considered. We may then

use this algorithm to ﬁnd the totally real quintic number ﬁeld(s) of smallest discriminant

satisfying some property. Hunter’s Theorem 10

3 Hunter’s Theorem

We discuss here the main theorem that will enable us to generate a table of all totally real quintic number ﬁelds (up to isomorphism) satisfying some ﬁeld discriminant bound. The result is due to J. Hunter and the reader is directed to [Hunter] and [Cohen2, Theorem

9.3.1] for more details. We will introduce Hunter’s Theorem and discuss how it may be used as a means to generate tables of number ﬁelds.

Given a number ﬁeld K of degree n satisfying |d(K)| < B for some B ∈ N, Hunter’s Theorem provides us with the existence of a nonrational algebraic integer of K with charac-

teristic polynomial having coeﬃcients that satisfy certain inequalities written only in terms

1 of n, B and a constant γn−1 dependent on n. The ﬁrst several values of this constant are given by

√ 2 1/3 3/5 2 γ1 = 1, γ2 = √ , γ3 = 2 , γ4 = 2, γ5 = 2 and γ6 = . 3 31/6

Theorem 3.1. (Hunter’s Theorem). Let K be a number ﬁeld of degree n over Q.

There exists α ∈ OK \Z with the following properties:

1. If σi(α) for i = 1, ..., n denote the algebraic conjugates of α in C and γn−1 is as given above, then n 1/(n−1) X 1 |d(K)| |σ (α)|2 ≤ Tr (α)2 + γ . i n K/Q n−1 n i=1

2. 0 ≤ Tr (α) ≤ n/2. K/Q

1Here we use Hermite’s constant, which is known explicitly only for n ≤ 8. See [Cohen, Proposition 6.4.1]. Hunter’s Theorem 11

In practice, Hunter’s Theorem is used as follows. Suppose a number ﬁeld K of degree n satisﬁes |d(K)| < B. Then there exists a nonrational algebraic integer α ∈ K satisfying

n 1/(n−1) 1/(n−1) X 1 |d(K)| n B |σ (α)|2 ≤ Tr (α)2 + γ ≤ + γ . i n K/Q n−1 n 4 n−1 n i=1

If we denote the characteristic polynomial of α by

n n n−1 n Y cα,Q(X) = X − a1X + ··· + (−1) an = (X − σi(α)), i=1

then we see that Hunter’s Theorem provides bounds for the coeﬃcients of cα,Q(X). Since we have a = Tr (α) we obtain immediately 0 ≤ a ≤ n/2. If we deﬁne 1 K/Q 1

a2 B 1/(n−1) t := 1 + γ 2 n n−1 n then Hunter’s Theorem gives n X 2 0 < |σi(α)| ≤ t2, i=1 and thus for each σi(α) we have 1/2 |σi(α)| < t2 .

Using the fact that the coeﬃcient ak is the kth elementary symmetric function of the σi(α), we obtain the bounds n |a | ≤ tk/2. k k 2

2 By the arithmetic-geometric mean inequality on |σi(α)| we have

n !1/n n Y 1 X t2 |a |2/n = |σ (α)|2 ≤ |σ (α)|2 ≤ n i n i n i=1 i=1

n/2 n/2 which gives a better bound |an| ≤ t2 /n for the constant term an. Hunter’s Theorem 12

Thus, given n, B ∈ N, Hunter’s Theorem allows us to create a list of polynomials such that every number ﬁeld K of degree n satisfying |d(K)| < B will contain a nonrational algebraic integer whose characteristic polynomial is in the list. For example, every number

field of degree 4 with absolute field discriminant less than 2.0×104 will contain an algebraic integer α whose characteristic polynomial cα,Q(X) satisfies the following properties:

4 3 2 1. cα,Q(X) = X − a1X + a2X − a3X + a4.

2. 0 ≤ a1 ≤ 2.

3. |a2| ≤ 135.

4. |a3| ≤ 428.

5. |a4| ≤ 31.

However, for extensions of composite degree such as the example just given, note that

a problem arises when we realize that the element α given by Hunter’s Theorem need

not be primitive. That is, α may generate a nontrivial subﬁeld of K, or equivalently, the

characteristic polynomial of α may be reducible over Q as a product of irreducible nonlinear terms. In such a situation, we must construct K by considering the relevant subﬁelds. For

more information on this process, see [Pohst]. As this problem never arises in ﬁelds of prime

degree, we do not consider it in detail here.

As a ﬁrst step to generating a table of all number ﬁelds of degree n satisfying |d(K)| < B,

we use Hunter’s Theorem to determine bounds for the coeﬃcients ak and create a list of all

polynomials in Z[X] satisfying these bounds. We then remove any polynomial from the list

that contains a linear factor in Q[X]. For the remaining reducible polynomials in the list, we must construct number ﬁelds of degree n from subﬁelds as noted above. The roots of

any irreducible polynomial in the list give a number ﬁeld of desired degree. Hunter’s Theorem 13

As a second step, we throw out those polynomials that give number fields outside the desired discriminant bounds. While our list of polynomials includes at least one characteristic polynomial of an algebraic integer inside each field satisfying the desired discriminant bounds, typically many fields outside these bounds will also be given. In the case that the roots of a polynomial generate a number field K such that |d(K)| ≥ B, we remove the polynomial from our list.

Thirdly, note that it is possible that multiple polynomials in our list may generate the same number field. To avoid including a number field more than once in our table, we must check for field isomorphisms. This can be done in several ways. When the computed table of number fields is relatively small, we can take any two fields in the table that have the same

ﬁeld discriminant and attempt to ﬁnd a bijection between them. For large tables of number

fields, this process is generally inefficient. A different approach is to apply a reduction algorithm such as POLRED ([Cohen, Algorithm 4.4.11]) to the list of polynomials before number field data is calculated. This has the effect of creating a pseudo-canonical defining polynomial for each number field under consideration. We adopt this approach as we expect our table of totally real quintic number fields to be rather large, and we therefore discuss the application of this approach in more detail.

As an example, consider the irreducible polynomials

5 4 3 2 p1(X) = X − X − 15X − 19X − 8X − 1

5 4 3 2 and p2(X) = X − X − 4X + 3X + 3X − 1.

These polynomials generate isomorphic totally real quintic number fields of field discriminant 14, 641. For the purpose of selecting a single polynomial to represent the number field

(what we may call a “pseudo-canonical deﬁning polynomial”) when this situation arises, we Hunter’s Theorem 14

will make use of the following deﬁnition. Let

5 X j Tj(pk) := βi i=1 where βi is the ith root of the polynomial pk. Then, in our example we have

T1(p1) = 1,T1(p2) = 1,T2(p1) = 31 and T2(p2) = 9.

Of course, T1 gives the field trace of the roots of each polynomial inside the number fields they generate, and the reader should note the similarity between T2 and our previous definition of t2. Namely, letting pk(X) = cα,Q(X) gives

n n 1/(n−1) X X a2 B T (p ) := β2 = |σ (α)|2 ≤ t := 1 + γ . 2 k i i 2 n n−1 n i=1 i=1

We treat T2 as a norm on the polynomials in our list, although it is not a norm in the strict mathematical sense. For each member of the list, we wish to compute the polynomial of least T2-norm generating an isomorphic number field. This is accomplished by the POLRED algorithm listed above, which has a relatively high level of underlying complexity and so we have not discussed it in detail here.1 After creating a new list of polynomials that represent pseudo-canonical defining polynomials for each number field under consideration, we remove repeated polynomials from the list and therefore limit the amount of isomorphism checking that will be required once our table is created.

It is important to note that the pseudo-canonical deﬁning polynomials of our new list may potentially fail to satisfy the coeﬃcient bounds given by Hunter’s Theorem.

1This reduction can be performed in PARI with the command polredabs() Hunter’s Theorem 15

To summarize, we use Hunter’s Theorem to generate a list of polynomials that contains the characteristic polynomial of some nonrational integer for every field of degree n satisfying some pre-determined field discriminant bound. We then remove polynomials from this list based on reducibility criteria and reconstruct any number field that may be represented in our list only by its subfields. After applying a reduction algorithm to limit resulting field isomorphisms, we use the list of polynomials to construct our table of number fields. As a

ﬁnal step, we consider isomorphic ﬁelds that may have been missed in the reduction process. Generating Totally Real Quintic Number Fields 16

4 Generating Totally Real Quintic Number Fields

In the quintic case the situation is less complex since no nontrivial subfields exist, and therefore Hunter’s Theorem can be applied to the search for a primitive element α of all quintic number fields satisfying some field discriminant bound. (Of course, this is true for any prime degree.) We can therefore restate Hunter’s Theorem in the quintic case as follows.

Theorem 4.1. (Hunter’s Theorem for Quintic Number Fields.) Let K be a quintic number ﬁeld. There exists an element α ∈ K with the following properties:

1. K = Q(α).

2. If σi(α) for i = 1, ..., 5 denote the algebraic conjugates of α in C, then

5 1/4 X 1 4 · d(K) σ (α)2 ≤ Tr (α)2 + . i 5 K/Q 5 i=1

3. 0 ≤ Tr (α) ≤ 2. K/Q

Where before we used Hunter’s Theorem to determine bounds on the coeﬃcients of the characteristic polynomial cα,Q(X) (where it is possible that α is primitive in a subﬁeld of

K), we see now that the condition α ∈ OK \Z gives K = Q(α) and we therefore consider the same bounds on the minimal polynomial mα,Q(X). Computationally, this is an advantage when generating tables of quintics as we need only consider irreducible monic ﬁfth degree polynomials that satisfy the bounds given by Hunter’s Theorem in order to include every

ﬁeld in our table.

As discussed in the previous chapter, we begin by constructing a list of polynomials having coeﬃcients that satisfy the bounds provided by Hunter’s Theorem. Namely, if our Generating Totally Real Quintic Number Fields 17

desired discriminant bound is B, we consider all irreducible polynomials

5 4 3 2 mα,Q(X) = X − a1X + a2X − a3X + a4X − a5

where !k/2 5 5 4 4B 1/4 |a | ≤ tk/2 ≤ + k k 2 k 5 5 and 1/4 !5/2 t 5/2 4 + 4B |a | ≤ 2 ≤ 5 5 . 5 5 5

We realize rather quickly that the amount of computation involved makes this process rather inefficient. For example, consider the calculations required to generate all quintic number fields K (up to isomorphism) where |d(K)| < 14, 642 = B. This requires first that we solve the inequalities provided by Hunter’s Theorem. That is, we consider all polynomials

5 4 3 2 mα,Q(X) = X − a1X + a2X − a3X + a4X − a5 where Hunter’s Theorem gives the bounds

0 ≤ a1 ≤ 2, |a2| ≤ 112, |a3| ≤ 374, |a4| ≤ 627 and |a5| ≤ 7.

This results in a list of 9,517,449,375 polynomials. Many of these are reducible and can thus be removed from consideration. Of those that remain, we find that many polynomials generate isomorphic number fields. After removing multiple instances of number fields due to isomorphisms by using a reduction algorithm such as POLRED as discussed in the previous chapter, we find that only 139 quintic number fields have absolute discriminant less than 14,642. Moreover, only one of these fields is totally real. Generating Totally Real Quintic Number Fields 18

The situation is summarized in the following table.

Table 1: Quintic Number Fields K with |d(K)| < 14, 642 Signature # of Number Fields min(d(K)) max(d(K)) [1, 2] 116 1,609 14,357 [3, 1] 22 −14, 631 −4, 511 [5, 0] 1 14,641 14,641

It is clear that this approach is inefficient if we aim to generate many totally real quintic number fields. However, there are several improvements that can be made. First, we construct separately better lower and upper bounds for the coefficient a2.

Placing a Lower Bound on a2

If mα,Q(X) is as above, then the jth power sum of α is given by

5 X j Tj(mα,Q) := σi(α) . i=1

Newton’s formulas (under the condition a0 = 1) are then written

k X j−1 kak = (−1) ak−jTj(mα,Q). j=1

Note that T1(mα,Q) gives the ﬁeld trace of α in the number ﬁeld that it generates. Specif- ically, we have T1(mα,Q) = a1. In our consideration of Hunter’s Theorem we already have bounds corresponding to the 2nd power sum of α. Namely,

a2 4B 1/4 T (m ) ≤ t := 1 + . 2 α,Q 2 5 5 Generating Totally Real Quintic Number Fields 19

We therefore have

2 X j−1 2a2 = (−1) a2−jTj(mα,Q) j=1 2 = a1 − T2(mα,Q).

Thus, knowing from Hunter’s Theorem that 0 ≤ a1 ≤ 2 we have the bound

1/4 1/4 4 4B 2 B − 5 + 5 −T (m ) a2 − T (m ) − − = ≤ 2 α,Q ≤ 1 2 α,Q = a . 5 20 2 2 2 2

Placing an Upper Bound on a2

Notice that if we require the quintic number ﬁeld K to be totally real, then the algebraic

integer α ∈ K given by Hunter’s Theorem has a separable minimal polynomial with ﬁve real

roots. Also, the ﬁrst, second and third derivatives of the minimal polynomial are separable

with real roots. We therefore consider the minimal polynomial and derivatives

5 4 3 2 mα,Q(X) = X − a1X + a2X − a3X + a4X − a5, m0 (X) = 5X4 − 4a X3 + 3a X2 − 2a X + a , α,Q 1 2 3 4 m00 (X) = 20X3 − 12a X2 + 6a X − 2a , α,Q 1 2 3 and m000 (X) = 60X2 − 24a X + 6a . α,Q 1 2

000 2 Hence m (X) has two real roots if and only if a2 < 2a /5. Therefore, in the worst possible α,Q 1

case, we have a2 ≤ 1.

These new bounds on a2 applied to our search for all totally real quintic number ﬁelds K satisfying the discriminant bound d(K) < 14, 642 = B give

−5 ≤ a2 ≤ 1. Generating Totally Real Quintic Number Fields 20

That is, the list of polynomials that need to be checked is reduced from 9,517,449,375 to

253,798,650. In fact, we find that a similar process as that used to place an upper bound on a2 can be used on a3, a4 and a5. For these coefficients, we reject altogether the bounds given by Hunter’s Theorem as they are computationally inadequate. Instead, we concentrate on the derivatives of the minimal polynomial mα,Q(X) and bound the remaining coefficients algorithmically using some elementary algebra and calculus techniques. As we’re interested from this point on in only totally real quintic number fields, we require that the minimal polynomial mα,Q(X) of the element α given by Hunter’s Theorem and its first through third derivatives have only real roots.

Bounding a3 with Polynomial Discriminants

We know from Hunter’s Theorem, Newton’s formulas and the polynomial discriminant of the third derivative of mα,Q(X) that

2 B 1/4 2a2 0 ≤ a ≤ 2 and − − ≤ a < 1 . 1 5 20 2 5

Given some rational integers a1 and a2 satisfying the above bounds, which values of a3 ∈ Z 00 3 2 result in the polynomial m (X) = 20X − 12a1X + 6a2X − 2a3 having three real roots? α,Q The polynomial discriminant of m00 (X) is given by α,Q

D m00 = −1728 25a2 + (8a3 − 30a a )a + 10a3 − 3a2a2 . 3 α,Q 3 1 1 2 3 2 1 2

00 This is a quadratic equation in a3. We require that D3 m > 0, and thus solving α,Q 00 00 D3(m ) = 0 tells us that m (X) has three real roots provided α,Q α,Q

R − S R + S 3 3 < a < 3 3 25 3 25 Generating Totally Real Quintic Number Fields 21

where

q 3 6 4 2 2 3 R3 = −4a1 + 15a1a2 and S3 = 16a1 − 120a1a2 + 300a1a2 − 125a2.

Bounding a4 with Minima and Maxima

00 Given coeﬃcients a1, a2 and a3 such that m (X) has three real roots, we can bound α,Q a4 as follows. Let

q 3 2 2 3 3 2 R4 = −4a1 + 15a1a2 − 25a3 + 5 −3a1a2 + 10a2 + 8a1a3 − 30a1a2a3 + 25a3

2 where solving R4 = 0 in terms of a2 gives a2 = 2a1/5, which is outside the bounds given

above and thus R4 6= 0. By the cubic formula, if we let

2 S4 = −36a1 + 90a2

00 and denote the roots of m (X) by β1, β2 and β3, then we have α,Q

√ 1/3 √ a1 (1 − i 3)R4 (1 + i 3)S4 β1 = + − , 5 10 · 21/3 1/3 1/3 180 · 2 R4

1/3 a1 R4 S4 β2 = − + , 5 5 · 22/3 1/3 1/3 90 · 2 R4

√ 1/3 √ a1 (1 + i 3)R4 (1 − i 3)S4 and β3 = + − . 5 10 · 21/3 1/3 1/3 180 · 2 R4

Consider the polynomial

M (X) = m0 (X) − a = 5X4 − 4a X3 + 3a X2 − 2a X. 4 α,Q 4 1 2 3 Generating Totally Real Quintic Number Fields 22

Then min{βi} and max{βi} for i = 1, 2, 3 are local minima of M4(X) and the remaining

0 root βj is a local maximum. It follows that m (X) will have four real roots provided that α,Q

− bM4(βj)c ≤ a4 ≤ − dmax{M4(min{βi}),M4(max{βi})}e .

For example, if we have a1 = 1, a2 = −5 and a3 = −5, then the relative extrema of

M4(X) are given by

β1 ≈ −1.12618, β2 ≈ 1.41167 and β3 ≈ 0.31451.

Plugging these values in to M4(X), we ﬁnd how far it is possible to shift M4(X) vertically so that it continues to have four real roots. This can be seen in the following graph.

In this case, we have bM4(β3)c = 1 and dM4(β2)e = −7, giving

−1 ≤ a4 ≤ 7.

Bounding a5 with Minima and Maxima Generating Totally Real Quintic Number Fields 23

0 In the same fashion, given a1, a2, a3 and a4 such that m (X) has four real roots, α,Q

we can place bounds on a5 so that mα,Q(X) will have ﬁve real roots. We ﬁrst calculate the roots of m0 (X) by the quartic formula. To simplify our calculations, we make the α,Q following substitutions. Let

3 2 2 Q = 54a2 − 216a1a2a3 + 540a3 + 432a1a4 − 1080a2a4,

q 2 3 2 R5 = −4(9a2 − 24a1a3 + 60a4) + Q ,

1/3 S5 = (Q + R5) ,

1/3 2 T = 2 (3a2 − 8a1a3 + 20a4),

S T U = 5 + , 1/3 15 · 2 5S5

4a2 2a V = 1 − 2 , 25 5

64a3 48a a 16a and W = 1 − 1 2 + 3 . 125 25 5

The reader may verify that the condition that m0 (X) has four real roots implies that α,Q 1/3 0 R5 > 0 and (Q + R5) is nonzero and real. If the roots of m (X) are denoted β1, β2, α,Q Generating Totally Real Quintic Number Fields 24

β3 and β4, then the quartic formula gives

√ s a1 U + V 1 W β1 = − − −U + 2V − √ , 5 2 2 4 U + V

√ s a1 U + V 1 W β2 = − + −U + 2V − √ , 5 2 2 4 U + V

√ s a1 U + V 1 W β3 = + − −U + 2V + √ , 5 2 2 4 U + V

√ s a1 U + V 1 W and β4 = + + −U + 2V + √ . 5 2 2 4 U + V

The polynomial

5 4 3 2 M5(X) = mα,Q(X) + a5 = X − a1X + a2X − a3X + a4X

has local maxima at β1 and β3, while β2 and β4 give local minima. Evaluating M5(X) at

these roots we ﬁnd that mα,Q(X) has ﬁve real roots in the case that

dmax{M5(β2),M5(β4)}e ≤ a5 ≤ bmin{M5(β1),M5(β3)}c .

An Algorithm for Generating Totally Real Quintic Number Fields

We use what we know from Hunter’s Theorem to bound the coeﬃcient a1 and to place

a lower bound on a2. We use polynomial discriminants to place an upper bound on a2. Then, we construct bounds for the remaining coeﬃcients using polynomial discriminants

and basic calculus. This provides us with the ability to only consider totally real ﬁelds.

We can therefore state the procedure of generating all totally real quintic number ﬁelds

satisfying some discriminant bound with the following algorithm. Generating Totally Real Quintic Number Fields 25

Algorithm 4.1. Given B ∈ N, this algorithm creates a table of all totally real quintic number ﬁelds K (up to isomorphism) such that d(K) < B, where each ﬁeld is represented in the table once and only once.

1. Set bounds for a2, as given by

$ 1 % 2 B 4 − − ≤ a ≤ 1. 5 20 2

2. For each a2 in the given range, do steps 3 and 4.

3. Set bounds for a1, as given by 0 ≤ a1 ≤ 2.

4. For each a1 in the given range, do steps 5 and 6.

5. Set bounds for a3 by polynomial discriminants.

6. For each a3 in the given range, do steps 7 and 8.

7. Set bounds for a4 by root ﬁnding and local minima/maxima.

8. For each a4 in the given range, do steps 9 and 10.

9. Set bounds for a5 by root ﬁnding and local minima/maxima.

10. For each a5 in the given range, do steps 11 through 15.

11. If the resulting polynomial is reducible, skip past step 15.

12. (T2-TEST, see below)

3 • If the resulting polynomial has minimal T2 form having X coeﬃcient greater

than a2, then skip past step 15 (because the polynomial has already been generated at a previous step of the algorithm). Generating Totally Real Quintic Number Fields 26

• If the resulting polynomial has minimal T2 form having X3 coeﬃcient less than a2 but still within the bounds determined in step 1, then skip past step 15 (because

the polynomial will be considered another step in the algorithm).

13. If the minimal T2 form of the polynomial is already in our list, we need not consider it twice and thus we skip past step 15.

14. Generate number ﬁeld data for the given polynomial.

15. If the discriminant of the resulting number ﬁeld is less than B, log the result.

(All polynomials have now been generated.)

16. Check for isomorphisms among entries having the same ﬁeld discriminant, remov-

ing multiple ﬁelds. The resulting table gives every totally real quintic number ﬁeld

satisfying the discriminant bound B.

For step 12 in the above algorithm, we use POLRED to determine a “pseudo-canonical defining polynomial” for each number field generated, as discussed previously. The current implementation in the PARI/GP system under the polredabs() function will always return a defining polynomial with a1 equal to 0, 1 or 2. Thus, if the a2 coefficient of the defining

polynomial is greater than the a2 coeﬃcient of the polynomial on which POLRED was performed, then we need not log the result as this deﬁning polynomial will already be in

our list (since the algorithm iterates on a2, progressing negatively). However, note that it

may happen to be the case that the a2 coefficient of the defining polynomial is outside the bound $ 1 % 2 B 4 − − ≤ a , 5 20 2 in which case we must log the result as Algorithm 4.1 will terminate before reaching this point. If the a2 coefficient of the defining polynomial is within this bound and also less than Generating Totally Real Quintic Number Fields 27

the a2 coeﬃcient of the polynomial on which POLRED was performed, we need not log the result as the deﬁning polynomial will be found at another step in Algorithm 4.1, while it may not yet be included in our list. This test with respect to the T2-norm, which we refer to

as the T2-TEST algorithm, is not required, but in practice can be a highly eﬀective method for limiting ﬁeld isomorphisms in the resulting table. We introduce it here more formally.

Algorithm 4.2. (T2-TEST) Given an irreducible quintic P and absolute ﬁeld discrim-

inant bound B, this algorithm returns 0 if the polynomial has a minimal T2 form with a

greater coeﬃcient in its third degree term or if the minimal T2 form has a smaller third degree term coeﬃcient that is within the bounds decided from B and Hunter’s Theorem.

Else, the algorithm returns 1. (As implemented above in Algorithm 4.1, the polynomial is useful only when this algorithm returns 1, in which case we say the polynomial “passes”

T2-TEST.)

1. Input irreducible quintic P and absolute discriminant bound B.

2. Let P ∗ =POLRED(P ).

1 3 ∗ 2 B 4 3. If the coeﬃcient of the X term of P is greater than or equal to − 5 − 20 and P 6= P ∗, then return 0 and exit.

4. Return 1. Tables of Quintic Number Fields of Small Discriminant 28

5 Tables of Quintic Number Fields of Small Discriminant

Algorithm 4.1 was written for use in the PARI/GP1 environment developed by Henri Cohen and maintained by Karim Belabas. For the ﬁeld discriminant bound B = 2.0 × 105, we compute that Algorithm 4.1 must be executed over the interval −10 ≤ a2 ≤ 1. Doing so yields the following table of results.

Table 2: Results of Algorithm 4.1 for −10 ≤ a2 ≤ 1

a2 Quintics Irreducible Pass T2-TEST d(K) < 2.0E5 0 0 0 0 0 −1 1 0 0 0 −2 10 0 0 0 −3 47 3 1 1 −4 153 22 14 8 −5 414 113 79 12 −6 985 371 288 11 −7 2,134 1,033 802 1 −8 4,271 2,372 1,914 1 −9 7,999 5,034 4,046 0 −10 14,137 9,651 7,972 0 totals 30,151 18,599 15,116 34

The first column in the table represents the a2 coefficient of the generated polynomials before POLRED is applied. (It is thus possible that the polynomials in our final table will have different a2 coefficients.) The second column gives the number of totally real quintics having a2 as the coefficient on their third degree term, again before POLRED is applied. The third column gives the number of those quintics from column 2 that are irreducible.

Out of those, column 4 tells us how many pass T2-TEST. Out of those polynomials passing

T2-TEST, column 5 tells us how many both satisfy the required ﬁeld discriminant bounds and are unique in the sense that they will not be represented by another polynomial in our

1See: http://pari.math.u-bordeaux.fr/ Tables of Quintic Number Fields of Small Discriminant 29

table.

The algorithm finds that there are no occurrences of a specific field discriminant being represented more than once, and so we find that there are 34 unique totally real quintic number fields of discriminant less than 200,000. This is verified by the results found in

[Schwarz] as well as by the available data in the QaoS system2. We now continue to larger discriminant bounds and state a theorem that our algorithm is capable of proving.

Theorem 5.1. (Uniqueness of Fields of Small Discriminant). There is at most one unique totally real quintic number ﬁeld of given discriminant D if D < 1, 810, 969. For

D = 1, 810, 969 there are two nonisomorphic totally real quintic number ﬁelds having the same discriminant.

Proof: If we set the minimum bound for a2 to −18 in Algorithm 4.1, then we find that this will generate all totally real quintic number fields having field discriminant less than

1,919,026. (This reduces almost insigniﬁcantly to 1,919,025 via Stickelberger’s Theorem.)

We then obtain a table having 1,022 totally real quintic number fields, but we notice that only 1,020 field discriminant values are recorded. We find that the polynomials

X5 − 2X4 − 8X3 + 9X2 + 16X − 7 and X5 − X4 − 7X3 + 7X2 + 6X − 5 both generate ﬁelds of discriminant 1,810,969 that upon closer inspection are determined to not be isomorphic. We also ﬁnd that the polynomials

X5 − 10X3 − 7X2 + 6X + 1 and X5 − X4 − 7X3 + 6X2 + 9X − 6 generate non-isomorphic ﬁelds of discriminant 1,891,397. Since no disciminant less than

2Querying Algebraic Objects System, See http://qaos.math.tu-berlin.de/cgi-bin/qaos/qaos.scm. The reader should note that our generating polynomials differ from those in the QaoS system due to the specific nature of the T2-TEST algorithm. However, the represented fields are isomorphic. Tables of Quintic Number Fields of Small Discriminant 30

1,810,969 is recorded as the ﬁeld discriminant of two non-isomorphic number ﬁelds, the result is shown.

The data in the above theorem is again consistent with that found in [Schwarz] and

QaoS, while the specific defining polynomials for each number field in these sources are not identical (and need not be) to our results here. Finding Fields of Smallest Discriminant 31

6 Finding Fields of Smallest Discriminant

Now that we have an algorithm to generate all totally real quintic number fields satisfying a given field discriminant bound, we have the ability to modify this algorithm to conduct a search for the number field of smallest discriminant satisfying a given property. In this section, we will develop a modifed version of Algorithm 4.1 to this end.

We change Algorithm 4.1 in the following context. It should be capable of ﬁnding the

first instance (with respect to minimality of the field discriminant) of a totally real quintic number field satisfying a given property, however large the field’s discriminant may be.

In the case that no totally real quintic number field satisfies this property, the algorithm should run indefinitely. Secondly, once an instance of a field satisfying the property is found, we can then generate all totally real fields of smaller discriminant and verify in each field whether or not the property holds.

As currently written, Algorithm 4.1 takes the ﬁeld discriminant bound B as an input value. However, the only coeﬃcient bounded directly from B is a2. We thus remove the

lower bound from this coeﬃcient, initializing the algorithm with a2 equal to its upper bound

(which is independent of B) and then recursively subtract 1 from a2 until the algorithm returns a ﬁeld having the desired property.

Once the algorithm ﬁnds a ﬁeld satisfying the given property, the discriminant of this

ﬁeld is then used to place a lower bound on a2 and the algorithm continues until this lower bound is reached. The initial ﬁeld returned by the algorithm may or may not be the

ﬁeld of smallest discriminant satisfying the property, so completion of the algorithm until

the calculated lower bound for a2 is reached is essential. Moreover, this lower bound can be improved if a ﬁeld of smaller discriminant satisfying the property is discovered after the

initial ﬁeld, in which case we simply recalculate the lower bound for a2 from the discriminant Finding Fields of Smallest Discriminant 32

of the new ﬁeld. There are, however, several computational issues that arise when we make these changes to Algorithm 4.1.

The purpose of the new algorithm that we are considering will be to find the first instance of a number field satisfying a given property. Since T2-TEST functions under the assumption that a2 is bounded below, we will need to replace T2-TEST with a weaker algorithm that does not contain this assumption. The result of this modification will be that potentially more isomorphic fields will be considered. However, once an instance of the desired property is found, the full T2-TEST algorithm can be used since a lower bound for a2 is then introduced.

Modifying Algorithm 4.1

We change the ﬁrst two steps of Algorithm 4.1 to the following.

1 1. Set a bound for a2, as given by

a2 ≤ −1.

2. Starting with a2 = −1 and then recursing on a2 = a2 − 1, do steps 3 and 4.

These modified steps now guarantee that the algorithm will run indefinitely unless an instance of the given property is found. We also make a modification to step 12, replacing it with the following variant of the T2-TEST algorithm.

12. (T2-TESTWEAK) Else, if the resulting polynomial has minimal T2 form having a

greater a2 coeﬃcient, skip past step ...

That is, we reduce every generated polynomial to its pseudo-canonical deﬁning polynomial via POLRED, only disregarding it in the case that the algorithm has already generated

1 We observe that the algorithm returns no results for a2 > −1. Finding Fields of Smallest Discriminant 33

the pseudo-defining polynomial explicitly. In the case that the pseudo-canonical defining polynomial has a third degree term coefficient less than the current a2, we calculate whether or not the generated field has the desired property, although it is possible for the algorithm

to advance to another a2 value and calculate data for the same ﬁeld.

We also make the required changes to Algorithm 4.1 with respect to checking for the desired property instead of discriminant values. In addition, we need to add a step that introduces a lower a2 bound and strengthens that bound when additional ﬁelds satisfying the property are discovered. The resulting algorithm can then be stated as follows.

Algorithm 6.1. Given a property P , this algorithm ﬁnds the totally real quintic number

ﬁeld K of smallest ﬁeld discriminant satisfying P .

1. Set a bound for a2, as given by

a2 ≤ −1.

2. Starting with a2 = −1 and then recursing with a2 = a2 − 1, do steps 3 and 4.

3. Set bounds for a1, as given by 0 ≤ a1 ≤ 2.

4. For each a1 in the given range, do steps 5 and 6.

5. Set bounds for a3 by polynomial discriminants.

6. For each a3 in the given range, do steps 7 and 8.

7. Set bounds for a4 by root ﬁnding and local minima/maxima.

8. For each a4 in the given range, do steps 9 and 10.

9. Set bounds for a5 by root ﬁnding and local minima/maxima.

10. For each a5 in the given range, do steps 11 through 14. Finding Fields of Smallest Discriminant 34

11. If the resulting polynomial is reducible, discard it.

12. (T2-TESTWEAK) If the resulting polynomial has minimal T2 form having a greater

a2 coeﬃcient, skip past step 14.

13. Generate number ﬁeld data for the given polynomial.

14. If the number ﬁeld generated satisﬁes P , do the following.

(a) Log the result.

(b) Calculate the corresponding a2 lower bound, adding this bound as a condition on limiting the recursion of step 2.

(Reaching this step implies the algorithm above has terminated.)

15. Check for isomorphisms among entries having the same ﬁeld discriminant, removing

multiple ﬁelds. The ﬁeld(s) of smallest discriminant in the list is (are) the desired

result. Examples of Fields of Smallest Discriminant 35

7 Examples of Fields of Smallest Discriminant

Algorithm 6.1 was coded in the PARI/GP environment and the following results were found.

In all cases, the result was veriﬁed in QaoS.

Example: The smallest totally real quintic number ﬁeld having an abelian Galois group.

This example causes the algorithm to terminate rather quickly since the totally real quintic number ﬁeld of smallest discriminant (14,641) has this property. Its Galois group is

Z5. The algorithm terminates with a2 = −6.

Example: The second smallest totally real quintic number ﬁeld having an abelian Galois group.

This ﬁeld has discriminant 390,625 and is given by a root of

X5 − 10X3 − 5X2 + 10X − 1.

It also has Galois group Z5. The algorithm terminates with a2 = −13 in this case.

Example: The smallest totally real quintic number ﬁeld having class number 2.

When a2 = −7 the algorithm returns the ﬁeld generated by a root of

X5 − 2X4 − 7X3 + 8X2 + 5X − 1.

This ﬁeld has discriminant 10,940,453. The calculated lower bound for a2 is then given to be −28. However, other ﬁelds of smaller discriminant are found with class number 2.

Speciﬁcally, when a2 = −11 a ﬁeld of class number 2, generated by a root of

X5 − 11X3 − 9X2 + 14X + 9 Examples of Fields of Smallest Discriminant 36

and having discriminant 4,010,276 is found. The new lower a2 bound is calculated to be −22. No ﬁelds of class number 2 having discriminant less than 4,010,276 are found.

Example: The smallest totally real quintic number ﬁeld having regulator greater than

100 but less than 101.

When a2 = −7 we ﬁnd a ﬁeld of discriminant 5,582,896. This gives a lower bound of

−24 for a2. Several other ﬁelds are found, including the ﬁeld generated by a root of

X5 − X4 − 11X3 + 7X2 + 18X + 6

and having discriminant 1,633,440. This gives a new lower bound of −18 for a2 and no ﬁelds of smaller discriminant are found. Totally Real Quintic Number Fields of Minimal Signature Group Rank 37

8 Totally Real Quintic Number Fields of Minimal Signature

Group Rank

Using Algorithm 6.1, we find instances of totally real quintic number fields of all signature group ranks. Moreover, we find the smallest discriminant totally real quintic number fields of signature group ranks 3, 4 and 5.

The Smallest Discriminant Rank 5 Field.

The totally real quintic number ﬁeld of discriminant 14,641 (which is unique by Theorem

5.1.) is the smallest discriminant totally real quintic number ﬁeld of signature group rank 5.

In fact, as previously mentioned, it is the smallest discriminant totally real quintic number

ﬁeld. This is veriﬁed by the QaoS database. It is generated by a root of the polynomial

X5 − X4 − 4X3 + 3X2 + 3X − 1,

has class number 1 and Galois group Z5.

The Smallest Discriminant Rank 4 Field.

The ﬁeld generated by a root of the polynomial

X5 − 2X4 − 3X3 + 5X2 + X − 1 has signature group rank 4 and discriminant 36,497. It is the smallest discriminant totally real quintic number field with signature group rank 4. In fact, it is the totally real quintic number field with the third smallest discriminant, a fact that is also verified by the QaoS database. It has class number 1 and Galois group S5. Totally Real Quintic Number Fields of Minimal Signature Group Rank 38

The Smallest Discriminant Rank 3 Field.

One of the new results that we have found is the following. The ﬁeld generated by a root of the polynomial

X5 − 2X4 − 6X3 + 8X2 + 8X + 1 has discriminant 638,597 and is found to be the smallest discriminant totally real field with signature group rank 3. It has class number 1 and Galois group S5. (This is the example provided in chapter 2 where we calculated the fundamental units of the field and then constructed matrices M1 and M2.) There are 222 totally real quintic number fields (up to isomorphism) of smaller discriminant, all having signature group ranks 4 or 5 and all having

class number 1. Out of these 222 totally real ﬁelds, 2 have Galois group Z5 and one has

Galois group D5. The rest have Galois group S5.

An Example of a Rank 2 Field.

Another new result is the ﬁrst known instance of a rank 2 ﬁeld. The totally real quintic

number ﬁeld generated by a root α of the polynomial

X5 − X4 − 21X3 − 7X2 + 68X + 60 has signature group rank 2 and discriminant 52,315,684. The fundemental units of this ﬁeld Totally Real Quintic Number Fields of Minimal Signature Group Rank 39

are given by

1 3 15 23 13 = α4 − α3 − α2 + α + , 1 4 4 4 4 2

1 3 15 19 23 = α4 − α3 − α2 + α + , 2 4 4 4 4 2

1 1 15 9 19 = α4 − α3 − α2 + α + 3 4 4 4 4 2

1 5 3 17 7 and = α4 − α3 − α2 + α + 4 4 4 4 4 2

This gives M1 and M2 as

     −1 1 1 1 1   1 0 0 0 0               −1 −1 1 1 1   1 1 0 0 0              M1 =  −1 −1 1 1 1  and M2 =  1 1 0 0 0  .              −1 1 1 1 1   1 0 0 0 0              −1 1 1 1 1 1 0 0 0 0

This field has class number 2 and Galois group S5. This is the smallest discriminant rank 2 field found thus far, and it is possible that this field is the totally real quintic number

ﬁeld of smallest discriminant having signature group rank 2. To verify this, one would need

to execute Algorithm 6.1 to the point where a2 = −41 (as determined by using 52,315,684

as an upper discriminant bound). We have currently executed the algorithm to a2 = −34, generating 38,562 totally real quintic number ﬁelds of signature group rank 2. Totally Real Quintic Number Fields of Minimal Signature Group Rank 40

Another rank 2 ﬁeld of interest is the one generated by a root of

X5 − 2X4 − 40X3 + 94X2 + 136X + 1 with discriminant 11,272,403,936,149 and class number 48. All rank 2 fields observed thus far have class numbers divisible by 2. This is consistent with the result of Armitage-Fröhlich, see [Armitage], stating that the 2-rank of the class group of any rank 2 field is at least 1.

In fact, rank 2 representative fields can currently be given from our results for all even class numbers up to 22. The field given here is the highest class number totally real quintic number field of signature rank 2 we have found so far.

An Example of a Rank 1 Field.

Principal among our results is the ﬁrst known instance of a totally real quintic number

ﬁeld with a totally positive system of fundamental units. The totally real quintic number

ﬁeld generated by a root of

X5 − 2X4 − 32X3 + 41X2 + 220X − 289 has signature group rank. The fundemental units of this ﬁeld are given by

1 3 15 23 13 = α4 − α3 − α2 + α + , 1 4 4 4 4 2

1 3 15 19 23 = α4 − α3 − α2 + α + , 2 4 4 4 4 2

1 1 15 9 19 = α4 − α3 − α2 + α + 3 4 4 4 4 2

1 5 3 17 7 and = α4 − α3 − α2 + α + 4 4 4 4 4 2 Totally Real Quintic Number Fields of Minimal Signature Group Rank 41

This gives M1 and M2 as

     −1 1 1 1 1   1 0 0 0 0               −1 1 1 1 1   1 0 0 0 0              M1 =  −1 1 1 1 1  and M2 =  1 0 0 0 0  .              −1 1 1 1 1   1 0 0 0 0              −1 1 1 1 1 1 0 0 0 0

It has discriminant 405,673,292,473 and Galois group S5. This number ﬁeld has class number 4 (in fact, the class group has Z/2Z × Z/2Z as a subgroup), again consistent with the results of Armitage-Fr¨ohlich.

When using Algorithm 6.1, the first example of a totally real quintic number field of signature group rank 1 is returned when a2 = −28. In fact, two fields are returned at this stage in the algorithm. The situation is summarized in the following table, which represents the first 34 rounds (corresponding to the coefficient a2) of Algorithm 6.1. The total number of quintics that were generated per round are given, followed by the number of those quintics that were irreducible and satisfied the T2-TESTWEAK algorithm. Corresponding signature group rank numbers are provided in the remaining columns. It is important to note that

the ﬁnal round of isomorphism reduction is not yet represented in this table (i.e., the table

was generated before Algorithm 6.1 reached step 15) and so it is possible that a ﬁeld may

be included twice. Totally Real Quintic Number Fields of Minimal Signature Group Rank 42

Table 3: Results of Algorithm 6.1 for −34 ≤ a2 ≤ 0

a2 Quintics T2-TEST Rank 5 Rank 4 Rank 3 Rank 2 Rank 1 -1 1 0 0 0 0 0 0 -2 10 0 0 0 0 0 0 -3 47 3 2 1 0 0 0 -4 153 22 17 5 0 0 0 -5 414 91 50 40 1 0 0 -6 985 300 149 144 7 0 0 -7 2,134 826 375 429 22 0 0 -8 4,271 1,947 827 1,054 66 0 0 -9 7,999 4,100 1,664 2,275 161 0 0 -10 14,137 7,972 3,067 4,512 393 0 0 -11 23,827 14,346 5,218 8,329 799 0 0 -12 38,549 24,527 8,890 14,169 1,467 1 0 -13 60,215 39,809 14,072 23,105 2,629 3 0 -14 91,299 62,582 21,772 36,251 4,555 4 0 -15 134,818 94,720 32,154 55,190 7,365 11 0 -16 194,466 139,750 46,899 81,679 11,143 29 0 -17 274,693 201,064 65,958 118,185 16,880 41 0 -18 381,047 283,406 92,990 165,824 24,514 78 0 -19 519,699 391,254 126,003 229,901 35,240 110 0 -20 698,170 532,053 170,070 312,190 49,633 160 0 -21 925,232 711,453 224,628 419,022 67,558 245 0 -22 1,210,822 939,115 294,765 553,192 90,799 359 0 -23 1,566,431 1,224,099 378,791 722,462 122,345 501 0 -24 2,005,409 1,577,377 487,086 930,258 159,349 684 0 -25 2,542,425 2,010,774 615,490 1,188,771 205,631 882 0 -26 3,194,622 2,538,769 769,177 1,502,129 266,255 1,208 0 -27 3,980,667 3,178,073 957,532 1,882,836 336,150 1,555 0 -28 4,921,858 3,944,717 1,180,614 2,340,127 421,896 2,078 2 -29 6,041,825 4,859,628 1,443,249 2,883,872 529,836 2,671 0 -30 7,366,995 5,944,154 1,759,020 3,526,266 655,401 3,466 1 -31 8,925,777 7,222,766 2,130,679 4,289,867 797,960 4,259 1 -32 10,750,697 8,720,466 2,549,574 5,179,218 986,227 5,446 1 -33 12,876,654 10,471,149 3,060,851 6,223,574 1,180,113 6,609 2 -34 15,342,424 12,500,682 3,634,108 7,432,370 1,426,041 8,162 1 totals 84,098,772 67,641,994 20,075,722 40,127,241 7,400,436 38,562 8 Totally Real Quintic Number Fields of Minimal Signature Group Rank 43

To date, the following totally real quintic number ﬁelds of signature group rank 1 have been found.

Table 4: Totally Real Quintic Number Fields of Signature Group Rank 1 Quintic Discriminant Factorization X5 − 2X4 − 32X3 + 41X2 + 220X − 289 405,673,292,473 = 9103 · 44564791 X5 − 28X3 − 15X2 + 150X + 121 1,003,227,973,721 = 34819 · 28812659 X5 − 2X4 − 35X3 + 62X2 + 268X − 394 1,415,563,272,496 = 24 · 197 · 21163 · 21221 X5 − X4 − 28X3 − 3X2 + 126X + 39 1,659,966,168,069 = 33 · 11 · 41 · 1117 · 122041 X5 − 33X3 − 47X2 + 46X + 39 1,906,910,160,004 = 2 · 476727540001 X5 − X4 − 30X3 + 22X2 + 119X + 44 2,254,477,774,469 = 23 · 312 · 67 · 1522369 X5 − 2X4 − 34X3 + 87X2 + 61X − 24 2,434,439,489,293 = 72 · 49682438557 X5 − 2X4 − 31X3 + 65X2 + 79X − 121 5,999,659,274,177 = 2207113 · 2718329 X5 − 2X4 − 36X3 + 91X2 + 81X − 46 7,334,794,603,613 = 7607 · 29803 · 32353 X5 − 2X4 − 38X3 + 73X2 + 237X − 458 9,728,717,436,101 = 533821 · 18224681 X5 − X4 − 38X3 + 75X2 + 146X − 169 10,145,103,898,421 = 19 · 113 · 4725246343 X5 − 37X3 − 23X2 + 146X − 57 10,239,239,468,004 = 22 · 3 · 40213 · 21218759 X5 − 39X3 − 27X2 + 218X + 171 11,065,999,441,892 = 22 · 31 · 41 · 47 · 281 · 164809 X5 − 2X4 − 33X3 + 32X2 + 112X − 71 11,943,307,601,417 = 3089 · 6967 · 554959 X5 − X4 − 40X3 + 73X2 + 217X − 401 14,772,238,087,817 = 7 · 20327 · 103818553 X5 − X4 − 36X3 + 37X2 + 125X − 97 18,790,538,749,457 = 8171 · 2299662067 X5 − 43X3 − 3X2 + 162X − 25 23,106,668,289,956 = 22 · 89 · 5021 · 12926981 X5 − 2X4 − 40X3 + 90X2 + 68X − 141 29,659,048,781,797 = 172 · 19 · 5401392967 X5 − X4 − 39X3 + 58X2 + 188X − 57 30,410,791,494,300 = 22 · 35 · 52 · 7 · 17 · 47 · 223757 X5 − 43X3 − 5X2 + 262X − 49 169,942,443,923,524 = 22 · 42485610980881

The smallest discriminant given for a rank 1 ﬁeld thus far is 405,673,292,473. If this is the rank one ﬁeld of smallest discriminant then Algorithm 6.1 must continue until a2 = −378 to prove that this is the case. A power regression relating the number of quintics generated

2 by Algorithm 6.1 during round a2 has a correlation coeﬃcient r = 0.9978. Integrating the given power function

p(x) = 0.0663x5.4118 between 0 and 378 tells us that roughly 3.47×1015 quintics must be generated by Algorithm

6.1 in order to verify that the ﬁeld given above is the rank 1 ﬁeld of smallest discriminant. Conclusion 44

9 Conclusion

We have provided examples of totally real quintic number fields of all signature group ranks, including the first known instance of a rank 2 field and also the first known instance of a field with a totally positive system of fundamental units. In addition, we have located the totally real quintic number field of smallest discriminant for all higher signature group ranks. The numerical data we have generated tells us that, up to this point, we are generating roughly one totally real quintic number field of signature group rank 1 for every 8.5 million fields that we consider. Rank 2 fields currently account for 0.05% of those generated, while ranks

3, 4 and 5 account for approximately 11%, 59.5% and 29.5%, respectively. Ultimately, given suﬃcient computing resources, this work may be used as a signiﬁcant step to providing unit group heuristics. References 45

References

[Armitage] “Class Numbers and Unit Signatures,” J.V. Armitage and A Frohlich, Mathematika, 14 (1967), pp. 94-98.

[Cohen] A Course in Computational Algebraic Number Theory, H. Cohen, Graduate Texts in Mathematics, Vol. 138, Springer, New York, 1993.

[Cohen2] Advanced Topics in Computational Number Theory, H. Cohen, Graduate Tests in Mathematics, Vol. 193, Springer, New York, 1999.

[Hunter] “The Minimum Discriminant of QUintic Fields,” Glasgow Math. Assoc. 3 (1957), 57-67.

[Pohst] “On the Computation of Number Fields of Small Discriminants Including the Minimum Discriminants of Sixth Degree Fields,” M. Pohst, Journal of Number Theory, 14 (1982), pp. 99-117.

[Schwarz] “A Table of Quintic Number Fields,” A. Schwarz, M. Pohst, and F. Diaz Y Diaz, Mathematics of Computation, Volume 63, Number 207 (1994), pp. 361-376.