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Class Field Theory Andrew Kobin 2013-2015 Contents Contents Contents 0 Introduction 1 1 Algebraic Number Fields 2 1.1 Rings of Algebraic Integers . .2 1.2 Dedekind Domains . .4 1.3 Ramification of Primes . .8 1.4 The Decomposition and Inertia Groups . 12 1.5 Norms of Ideals . 15 1.6 Discriminant and Different . 17 1.7 The Class Group . 24 1.8 The Hilbert Class Field . 32 1.9 Orders . 42 1.10 Units in a Number Field . 49 2 Class Field Theory 58 2.1 Valuations and Completions . 58 2.2 Frobenius Automorphisms and the Artin Map . 65 2.3 Ray Class Groups . 69 2.4 L-series and Dirichlet Density . 75 2.5 The Frobenius Density Theorem . 83 2.6 The Second Fundamental Inequality . 89 2.7 The Artin Reciprocity Theorem . 94 2.8 The Conductor Theorem . 99 2.9 The Existence and Classification Theorems . 101 2.10 The Cebotarevˇ Density Theorem . 103 2.11 Ring Class Fields . 110 3 Quadratic Forms and n-Fermat Primes 115 3.1 Binary Quadratic Forms . 115 3.2 The Form Class Group . 119 3.3 n-Fermat Primes . 124 A Appendix 128 A.1 The Four Squares Theorem . 128 A.2 The Snake Lemma . 129 A.3 Cyclic Group Cohomology . 130 A.4 Helpful Magma Functions . 132 i 0 Introduction 0 Introduction These notes are a product of nearly two years of research in class field theory as part of my Master's thesis at Wake Forest University. The main topics covered are: Algebraic number fields and their extensions Factorization of primes in number field extensions The class group The Hilbert class field Dirichlet's unit theorem Valuations and completions Ray class groups Dirichlet L-series, Dirichlet density and the proof of Dirichlet's theorem on primes in arithmetic progression The main theorems of class field theory: { Artin reciprocity { The Conductor Theorem { The fundamental equality { The Existence and Classification Theorems An extended discussion of Frobenius' and Cebotarev'sˇ density theorems Ring class fields and orders Quadratic forms and n-Fermat primes In fact the first motivation for studying these topics is to fully answer the question, as described in [7], \Given a positive integer n, when can a prime number be written in the form x2 + ny2?" The reader will see that although the question has a rather elementary statement, it requires the depth and power of class field theory to fully understand. After describing the answer to this first question, we will turn our attention to the much more difficult, and unanswered question, \Given a positive integer n, if x2 + ny2 is prime, when is y2 + nx2 also prime?" In certain sections (1.8, 2.10 and 3.3) we use the Magma Computational Algebra System to handle large or complicated computations. Many of the basic commands can be found in the Magma handbook, available at http://magma.maths.usyd.edu.au/magma/handbook/ through the University of Sydney's Computational Algebra Group. 1 1 Algebraic Number Fields 1 Algebraic Number Fields In the first chapter we provide a detailed description of the main topics in algebraic number theory: algebraic number fields, rings of integers, the behavior of prime ideals in extensions, norms of ideals, the discriminant and different, the class group, the Hilbert class field, orders and Dirichlet's unit theorem. 1.1 Rings of Algebraic Integers Let Q be an algebraic closure of Q. Then Q is an infinite dimensional Q-vector space and every polynomial f 2 Q[x] splits in Q[x]. An example of such an algebraic closure is Q = fu 2 C j f(u) = 0 for some f 2 Q[x]g. Then Q ⊂ Q ⊂ C. Note that any two choices of Q are isomorphic. One of the most important elements of a number field we will be working with is: Definition. An element α 2 Q is an algebraic integer if it is a root of some monic polynomial with coefficients in Z. p 2 1 Example 1.1.1. 2 is an algebraic integer since it is a root of x − 2. However, 2 ; π and 1 e are not algebraic integers. We will see in a moment why 2 is not algebraic, but the proof for π and e is famously difficult. Note that the set of algebraic integers in Q is precisely the integers Z. In a moment we will generalize this set to fields other than Q. Definition. The minimal polynomial of α 2 Q is the monic polynomial f 2 Q[x] of minimal degree such that f(α) = 0. The minimal polynomial of α is unique, as the following lemma shows. Lemma 1.1.2. Suppose α 2 Q. Then the minimal polynomial f of α divides any other polynomial h such that h(α) = 0. Proof. Suppose h(α) = 0. Then by the division algorithm, h = fq + r with deg r < deg f. Note that r(α) = h(α) − f(α)q(α) = 0 so α is a root of r. But since deg f is minimal among all polynomials of which α is a root, r must be 0. This shows that f divides h. Lemma 1.1.3. If α 2 Q is an algebraic integer then the minimal polynomial has coefficients in Z. Proof. Let f 2 Q[x] be the minimal polynomial of α. Since α is an algebraic integer, there is some g 2 Z[x] such that g(α) = 0. By Lemma 1.1.2, g = fh for some monic h 2 Q[x]. Suppose f 62 Z[x]. Then there is some prime p dividing the denominator of at least one of the coefficients of f; let pi be the largest power of p that divides a denominator. Likewise let pj be the largest power of p that divides the denominator of a coefficient of h. Then pi+jg = (pif)(pjh) and reducing mod p gives 0 on the left, but two nonzero polynomials in Fp[x] on the right, a contradiction. Hence f 2 Z[x]. 2 1.1 Rings of Algebraic Integers 1 Algebraic Number Fields An important characterization of algebraic integers is proven in the next proposition. Pn i Proposition 1.1.4. α 2 Q is an algebraic integer if and only if Z[α] = f i=0 ciα : ci 2 Z; n ≥ 0g is a finitely generated Z-module. Proof. ( =) ) Suppose α is integral with minimal polynomial f 2 Z[x], where deg f = k. Then Z[α] is generated by 1; α; : : : ; αk−1. ( ) = ) Suppose α 2 Q and Z[α] is generated by f1(α); : : : ; fn(α). Let d ≥ M where M = maxfdeg fi j 1 ≤ i ≤ ng. Then n d X α = aifi(α) i=1 n d X for some choice of ai 2 Z. Hence α is a root of x − aifi(x) so it is integral. i=1 1 1 Example 1.1.5. α = 2 is not an algebraic integer since Z 2 is not finitely generated as a Z-module. Definition. For a given algebraic closure Q of Q, we will denote the set of all algebraic integers in Q by Z. This set inherits the binary operations + and · from Q, and an important property is that Z is closed under these operations: Proposition 1.1.6. The set Z of all algebraic integers is a ring. Proof. Note that 0 is a root of the zero polynomial, so 0 2 Z. Then it suffices to prove closure under addition and multiplication. Suppose α; β 2 Z and let m and n be the degrees of their respective minimal polynomial. Then 1; α; : : : ; αm−1 span Z[α] and 1; β; : : : ; βn−1 likewise span Z[β]. So the elements αiβj for 1 ≤ i ≤ m; 1 ≤ j ≤ n span Z[α; β], so this Z-module is finitely generated. This implies that the submodules Z[α + β] and Z[αβ] of Z[α; β] are also finitely generated, so it follows by Proposition 1.1.4 that α + β and αβ are algebraic integers. The two most important objects of study in algebraic number theory are number fields and their associated rings of integers, which are defined below. Definition. A number field is a subfield K ⊂ Q such that K is a finite dimensional vector space over Q. The dimension of K=Q is called the degree of the field extension, denoted [K : Q]. Definition. The ring of integers of a number field K is OK = K \ Z = fα 2 K j α is an algebraic integerg: Example 1.1.7. Q is the unique number field of degree 1, and its ring of integers is the rational integers Z. 3 1.2 Dedekind Domains 1 Algebraic Number Fields Example 1.1.8. Q(i) is a number field of degree 2. Its ring of integers is Z[i], the Gaussian integers. p p Example 1.1.9. K = Q( p5) has ring of integers OK = Z[(1 + 5)=2]. The reader may recognize this number (1 + 5)=2 as the golden ratio. An object we will study in Section 1.9 is: Definition. An order in OK is any subring O ⊂ OK such that the quotient OK =O of abelian groups is finite. Example 1.1.10. For OQ(i) = Z[i], the subring Z+niZ is an order for every nonzero n 2 Z. However, Z ⊂ Z[i] is not an order since Z does not have finite index in Z[i]. Example 1.1.11. For K = Q(α) where α is an algebraic integer, Z[α] is an order in OK but in general Z[α] 6= OK .
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