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Inverse Galois Problem for Totally Real Number Fields

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Inverse Galois Problem for Totally Real Number Fields Cornell University Mathematics Department Senior Thesis Inverse Galois Problem for Totally Real Number Fields Sudesh Kalyanswamy Class of 2012 April, 2012 Advisor: Prof. Ravi Ramakrishna, Department of Mathematics Abstract In this thesis we investigate a variant of the Inverse Galois Problem. Namely, given a finite group G, the goal is to find a totally real extension K=Q, neces- sarily finite, such that Gal(K=Q) is isomorphic to G. Questions regarding the factoring of primes in these extensions also arise, and we address these where possible. The first portion of this thesis is dedicated to proving and developing the requisite algebraic number theory. We then prove the existence of totally real extensions in the cases where G is abelian and where G = Sn for some n ≥ 2. In both cases, some explicit polynomials with Galois group G are provided. We obtain the existence of totally real G-extensions of Q for all groups of odd order using a theorem of Shafarevich, and also outline a method to obtain totally real number fields with Galois group D2p, where p is an odd prime. In the abelian setting, we consider the factorization of primes of Z in the con- structed totally real extensions. We prove that the primes 2 and 5 each split in infinitely many totally real Z=3Z-extensions, and, more generally, that for primes p and q, p will split in infinitely many Z=qZ-extensions of Q. Acknowledgements I would like to thank Professor Ravi Ramakrishna for all his guidance and support throughout the year, as well as for the time he put into reading and editing the thesis. I would also like to thank the Hunter R. Rawlings III Cornell Presidential Research Scholars Program for their financial support over the last four years. Contents 1 Introduction 3 2 Algebraic Number Theory 5 2.1 Introduction to Number Fields . 5 2.1.1 Definition and Embeddings . 5 2.1.2 Norm and Trace . 7 2.2 Dedekind Domains and Ideal Factorization . 14 2.2.1 Localization . 14 2.2.2 Noetherian Rings and Modules . 16 2.2.3 Dedekind Rings . 21 2.2.4 Unique Factorization into Prime Ideals . 22 2.3 Ring of Integers . 27 2.3.1 Definition and Example . 27 2.3.2 Proof OK is a ring . 30 2.3.3 Extension of Dedekind Rings . 32 2.3.4 Discriminant . 37 2.3.5 Factoring Prime Ideals in Extensions . 41 2.3.6 Ramified Primes . 55 2.3.7 Decomposition and Inertia Groups . 61 2.3.8 Artin Automorphism . 73 3 Finite abelian case 77 3.1 Existence . 77 3.1.1 Four Essential Theorems . 77 3.1.2 The Proof . 80 3.2 Primes in Extensions . 83 3.2.1 Ramified Primes . 84 3.2.2 Dirichlet Density . 87 3.2.3 Primes which split completely . 89 3.2.4 Other Primes . 93 3.3 Polynomials with Abelian Galois group . 96 3.4 Splitting of primes 2 and 5 in Z=3Z-extensions . 101 3.4.1 Motivation . 101 1 3.4.2 Cubic Character . 102 3.4.3 Proof that 2 and 5 split in infinitely many Z=3Z extensions 106 3.5 Splitting of p in Z=qZ-extensions . 110 4 Symmetric Groups 112 4.1 Additional Background Material . 112 4.1.1 p-adic Numbers . 112 4.1.2 Approximation Theorem . 118 4.1.3 Alternate Method for Computing Galois Groups . 120 4.2 Realizing Sn as Galois group of totally real number field . 122 4.3 Explicit Polynomials: Symmetric Group . 125 5 Other Groups 127 5.1 Dihedral Groups . 127 5.1.1 Class Group . 127 5.1.2 Hilbert Class Field . 129 5.1.3 D2p, p prime . 130 5.2 Groups of Odd Order . 132 Appendices 134 A Fields and Galois Theory 134 A.1 Fields and Field Extensions . 134 A.2 Testing Irreducibility of Polynomials . 136 A.3 Galois Theory . 137 A.4 Composite Extensions . 142 B Galois Theory of Finite Fields 146 B.1 Existence and Uniqueness . 146 B.2 Normal and Separable . 146 B.3 Galois group . 147 C Cyclotomic Fields 149 D Explicit Polynomials: Abelian Case 151 E Examples: Dihedral Groups 154 2 Chapter 1 Introduction This thesis investigates a variant of the Inverse Galois Problem, which is an open problem in number theory. Given a polynomial or a field extension, one could calculate its Galois group, and though this can be difficult, especially for poly- nomials of large degree, it is \doable." Different methods for calculating Galois groups are used continuously throughout this thesis, so those not familiar with these methods (or terms) should consult Appendices A-C. The Inverse Galois Problem is Galois theory \in reverse." Namely, given a fi- nite group G, the question is whether G occurs as a Galois group of some (finite) extension of Q. If Q is not required to be the base field, then it is a theorem that every finite group occurs as the Galois group of some finite extension of fields. However, over Q, the problem is not yet solved. And even for groups for which the problem is solved, another difficulty is actually producing a polynomial in Q[x] with the prescribed Galois group. This is not to say that no progress has been made. For example, the follow- ing theorem of Shafarevich solves the problem for a whole class of groups, from [10]. Theorem 1.0.1. If G is a finite solvable group, then G occurs as a Galois group over Q. However, the proof Shafarevich provided did not include explicit polynomials. In this thesis, the problem is amended slightly by adding another restriction to the possible fields. Not only do we want our fields to be finite extensions of Q, we require that they be totally real, which we will define in the next chapter. As expected, this makes the problem slightly more challenging for some classes of groups. Chapter 2 serves as an introduction to most of the algebraic number theory 3 needed for this thesis, including the ring of integers, factorizations of ideals in extensions, decomposition and inertia groups, and the Frobenius automorphism. Chapter 3 addresses the abelian case. We prove that all finite abelian groups occur as Galois groups of totally real fields, and also study how primes of Z factor in these extensions. For example, we calculate the Dirichlet density of the primes which split completely in these totally real extensions. These terms are defined in chapters 2 and 3. We begin chapter 4 by providing a brief introduction to p-adic numbers, which we then use to prove that Sn can be realized as a Galois group of a totally real number field. Lastly, we outline a method to obtain D2p as a Galois group over Q, again of a totally real field, in chapter 5. Appendices A-C cover most of the definitions and theorems from Field Theory and Galois Theory which are required to understand the material in the main portion of the thesis. 4 Chapter 2 Algebraic Number Theory In this chapter, we introduce most of the algebraic number theory needed for this thesis. The primary motivation for writing this chapter is to provide as self- contained a thesis as possible, but almost all of these theorems can be found in other books, and we cite them along the way. Of course, there will be some results which we will not have the time to prove, but we state them and refer to them as needed. 2.1 Introduction to Number Fields 2.1.1 Definition and Embeddings This thesis is concerned with totally real number fields. In this section, the goal is to explain what these are, as well as introduce the norm and the trace. We will be following the beginning of [6]. Definition 2.1.1. A number field K is a finite extension of Q. The dimension dimQ K of K as a vector space over Q is called the degree of the number field, and is denoted [K : Q]. p For example, K = Q( 2) is a number field of degree two and L = Q(21=3) is a number field of degree three. It is important to note that a number field is not necessarily a Galois extension of Q, as the second example shows, but many of the results we will prove become considerably nicer (or easier) when dealing with Galois extensions. All number fields are simple extensions of Q. That is, a number field L has the form L = Q(α) for some element α 2 L. This follows from the following theorem, which is only stated. A proof can be found in [5], page 595. Theorem 2.1.2 (Primitive Element Theorem). If K=F is a finite, separable field extension, then K=F is a simple extension. 5 Since Q has characteristic zero, all finite extensions are separable (Theorem A.3.5), and the theorem applies to all number fields. There do exist extensions which are not simple, but by the above reasoning they necessarily have charac- teristic p for some prime p. This theorem gives us a useful way of working with number fields since they can be generated by one element. Now suppose L = Q(α) is a number field of degree n. Since C=Q is algebraically closed, L can be regarded as a subfield of C.p But there could be several ways of embedding L into C. For example, if L = Q(p 2), then anp embeddingp (i.e. an in- jective homomorphism)p σ : L,! C can map 2 to either 2 or − 2.
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