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

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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 comput- ing 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 num- ber 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.
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