ALGEBRAIC NUMBER THEORY Romyar Sharifi Contents Introduction 7 Part 1. Algebraic number theory 9 Chapter 1. Abstract algebra 11 1.1. Tensor products of fields 11 1.2. Integral extensions 13 1.3. Norm and trace 18 1.4. Discriminants 22 1.5. Normal bases 29 Chapter 2. Dedekind domains 33 2.1. Fractional ideals 33 2.2. Dedekind domains 34 2.3. Discrete valuation rings 39 2.4. Orders 43 2.5. Ramification of primes 47 2.6. Decomposition groups 52 Chapter 3. Applications 57 3.1. Cyclotomic fields 57 3.2. Quadratic reciprocity 61 3.3. Fermat’s last theorem 63 Chapter 4. Geometry of numbers 67 4.1. Lattices 67 4.2. Real and complex embeddings 70 4.3. Finiteness of the class group 71 4.4. Dirichlet’s unit theorem 74 Chapter 5. Valuations and completions 79 5.1. Global fields 79 5.2. Valuations 80 3 4 CONTENTS 5.3. Completions 88 5.4. Extension of valuations 97 5.5. Local fields 101 Chapter 6. Local-global theory 105 6.1. Semi-local theory 105 6.2. Ramification revisited 110 6.3. Differents and discriminants 113 Part 2. Cohomology 121 Chapter 7. Group cohomology 123 7.1. Group rings 123 7.2. Group cohomology via cochains 124 7.3. Group cohomology via projective resolutions 129 7.4. Homology of groups 132 7.5. Induced modules 134 7.6. Tate cohomology 136 7.7. Dimension shifting 141 7.8. Comparing cohomology groups 142 7.9. Cup products 152 7.10. Tate cohomology of cyclic groups 159 7.11. Cohomological triviality 162 7.12. Tate’s theorem 166 Chapter 8. Galois cohomology 171 8.1. Profinite groups 171 8.2. Cohomology of profinite groups 178 8.3. Galois theory of infinite extensions 182 8.4. Galois cohomology 185 8.5. Kummer theory 187 Part 3. Class field theory 193 Chapter 9. Nonarchimedean local fields 195 9.1. Multiplicative groups 195 9.2. Tamely ramified extensions 198 9.3. Ramification groups 201 Chapter 10. Class formations 209 CONTENTS 5 10.1. Reciprocity maps 209 10.2. Norm groups 215 10.3. Class field theory over finite fields 219 Chapter 11. Local class field theory 223 11.1. The Brauer group of a local field 223 11.2. Local reciprocity 228 11.3. Norm residue symbols 230 11.4. The existence theorem 235 11.5. Class field theory over Qp 239 11.6. Ramification groups and the unit filtration 243 Chapter 12. Global class field theory via ideals 247 12.1. Ray class groups 247 12.2. Statements 250 12.3. Class field theory over Q 256 12.4. The Hilbert class field 258 Chapter 13. Global class field theory via ideles` 261 13.1. Restricted topological products 261 13.2. Adeles 263 13.3. Ideles` 269 13.4. Statements 273 13.5. Comparison of the approaches 276 13.6. Cohomology of the ideles` 280 13.7. The first inequality 282 13.8. The second inequality 284 13.9. The reciprocity law 288 13.10. Power reciprocity laws 294 Introduction At its core, the ancient subject of number theory is concerned with the arithmetic of the integers. The Fundamental Theorem of Arithmetic, which states that every positive integer factors uniquely into a product of prime numbers, was contained in Euclid’s Elements, as was the infinitude of the set of prime numbers. Over the centuries, number theory grew immensely as a subject, and techniques were developed for approaching number-theoretic problems of a various natures. For instance, unique factorization may be viewed as a ring-theoretic property of Z, while Euler used analysis in his own proof that the set of primes is infinite, exhibiting the divergence of the infinite sum of the reciprocals of all primes. Algebraic number theory distinguishes itself within number theory by its use of techniques from abstract algebra to approach problems of a number-theoretic nature. It is also often considered, for this reason, as a subfield of algebra. The overriding concern of algebraic number theory is the study of the finite field extensions of Q, which are known as number fields, and their rings of integers, analogous to Z. The ring of integers O of a number field F is the subring of F consisting of all roots of all monic polynomials in [x]. Unlike , not all integer rings are UFDs, as one sees for instance by considering Z Z p the factorization of 6 in the ring Z[ −5]. However, they are what are known as Dedekind domains, which have the particularly nice property that every nonzero ideal factors uniquely as a product of nonzero prime ideals, which are all in fact maximal. In essence, prime ideals play the role in O that prime numbers do in Z. A Dedekind domain is a UFD if and only if it is a PID. The class group of a Dedekind domain is roughly the quotient of its set of nonzero ideals by its nonzero principal ideals, and it thereby serves as something of a measure of how far a Dedekind domain is from being a prinicipal ideal domain. The class group of a number field is finite, and the classical proof of this is in fact a bit of analysis. This should not be viewed as an anomalous encroachment: algebraic number theory draws heavily from the areas it needs to tackle the problems it considers, and analysis and geometry play important roles in the modern theory. Given a prime ideal p in the integer ring O of a number field F, one can define a metric on F that measures the highest power of p dividing the difference of two points in F. If the finite field O=p has characteristic p, then the completion Fp of F with respect to this metric is known as a p-adic field, and the subring Op that is the completion of O is called its valuation ring. In the case that F = Q, 7 8 INTRODUCTION one obtains the p-adic numbers Qp and p-adic integers Zp. The archimedean fields R and C are also completions of number fields with respect to the more familiar Euclidean metrics, and are in that sense similar to p-adic fields, but the geometry of p-adic fields is entirely different. For instance, a sequence of integers converges to 0 in Zp if and only it is eventually congruent to zero modulo arbitrarily high powers of p. It is often easier to work with p-adic fields, as solutions to polynomial equations can be found in them by successive approximation modulo increasing powers of a prime ideal. The “Hasse principle” asserts that the existence of a solution to polynomial equations in a number field should be equivalent to the existence of a solution in every completion of it. (The Hasse principle does not actually hold in such generality, which partially explains the terminology.) Much of the formalism in the theory of number fields carries over to a class of fields of finite characteristic, known as function fields. The function fields we consider are the finite extensions of the fields of rational functions Fp(t) in a single indeterminate t, for some prime p. Their “rings of integers”, such as Fp[t] in the case of Fp(t), are again Dedekind domains. Since function fields play a central role in algebraic geometry, the ties here with geometry are much closer, and often help to provide intuition in the number field case. For instance, instead of the class group, one usually considers the related Picard group of divisors of degree 0 modulo principal divsors. The completions of function fields are fields of Laurent series over finite fields. We use the term “global field” refer to number fields and function fields in general, while the term “local field” refers to their nonarchimedean completions. An introductory course in algebraic number theory can only hope to touch on a minute but essen- tial fraction of the theory as it is today. Much more of this beautiful edifice can be seen in some of the great accomplishments in the number theory of recent decades. Chief among them, of course, is the proof of Fermat’s last theorem, the statement of which is surely familiar to you. Wiles’ proof of FLT is actually rather round-about. It proceeds first by showing that a certain rational elliptic curve that can be constructed out of a solution to Fermat’s equation is not modular, and then that all (or really, enough) rational elliptic curves are modular. In this latter aspect of the proof are contained advanced methods in the theory of Galois representations, modular forms, abelian varieties, deformation theory, Iwasawa theory, and commutative ring theory, none of which we will be able to discuss. NOTATION 0.0.1. Throughout these notes, we will use the term ring to refer more specifically to a nonzero ring with unity. Part 1 Algebraic number theory CHAPTER 1 Abstract algebra In this chapter, we introduce the many of the purely algebraic results that play a major role in algebraic number theory, pausing only briefly to dwell on number-theoretic examples. When we do pause, we will need the definition of the objects of primary interest in these notes, so we make this definition here at the start. DEFINITION 1.0.1.A number field (or algebraic number field) is a finite field extension of Q. We have the following names for extensions of Q of various degrees. DEFINITION 1.0.2.A quadratic (resp., cubic, quartic, quintic, ...) field is a degree 2 (resp., 3, 4, 5, ...) extension of Q.