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INTRODUCTION to CLASS FIELD THEORY Contents 1. Theorems Of INTRODUCTION TO CLASS FIELD THEORY BOWEN WANG Abstract. This paper introduces basic theorems of class field theory and discusses Hilbert class fields. As an application of Hilbert class field, a partial solution to the problem \primes of the form p = x2 + ny2" is given. Contents 1. Theorems of Class Field Theory 1 2. Solution of p = x2 + ny2 for infinitely many n 7 Appendix A. A Quadratic Field Example 10 Acknowledgments 11 References 11 1. Theorems of Class Field Theory In this section we will introduce some theorems of class field theory and discuss the Hilbert class field of a number field. In order to discuss class field theory, we first define the unramified extension: Definition 1.1. Prime ideals of OK are called finite primes to distinguish them from infinite primes, which are embeddings of K into C. A real infinite prime is an embedding σ : K −! R, while a complex infinite prime is a pair of complex conjugate embeddings σ; σ¯ : K −! C; σ 6=σ ¯. Definition 1.2. Let K ⊂ L be a finite extension. Then, an infinite prime σ of K ramifies in L if σ is real but has an extension to L which is complex. An extension K ⊂ L is unramified if it is unramified at all primes, finite or infinite. Intuitively, the Hilbert class field of a number field K is the maximal unramified extension of K (a formal definition will be given later). However, the existence of Hilbert class fields is nontrivial and we need to develop some class field theory in order to show it. We first introduce the notion of a modulus, which is important in theorems of class field theory. Definition 1.3. Given a number field K, a modulus in K is a formal product Y m = pnp p over all primes p, finite or infinite, of K, where the exponents must satisfy: (i) np ≥ 0, and at most finitely many are nonzero. (ii) np = 0 if p is a complex infinite prime. 1 2 BOWEN WANG (iii) np ≤ 1 if p is a real infinite prime. A modulus m may thus be written as m0m1, where m0 is an OK -ideal and m1 is a product of distinct real infinite primes of K. When all of the exponents np = 0, we set m = 1. Note that for a purely imaginary field K, a modulus is simply an ideal of OK . We can now generalize the idea of ideal class group using the modulus m. Definition 1.4. Given a modulus m, let IK (m) be the group of all fractional OK - ideals relatively prime to m, and let PK;1(m) be the subgroup of IK (m) generated by the principal ideals αOK , where α 2 OK satisfies α ≡ 1 mod m and σ(α) > 0 for every real infinite prime σ dividing m1 Definition 1.5. A subgroup H ⊂ IK (m) is called a congruence subgroup for m if it satisfies PK;1(m) ⊂ H ⊂ IK (m) and the quotient IK (m)=H is called a generalized ideal class group for m. Notice that if we take m = 1, then PK = PK;1(1) is a congruence subgroup. Therefore the ideal class group Cl(OK ) is a generalized ideal class group. The main idea of class field theory, as we shall see, is the correspondence between the generalized ideal class group and the Galois group of abelian extensions of K, which are linked via the Artin map. Therefore we will now discuss the Artin symbol and the Artin map to see the isomorphism which defines the Hilbert class field. Lemma 1.6. Let K ⊂ L be a Galois extension, and let P be a prime of OK which is unramified in L. If P is a prime of OL containing P , then there is a unique element σ 2 Gal(L=K) such that for all α 2 OL, σ(α) ≡ αN(P ) mod P; where N(P ) = jOK =P j is the norm of P . Proof. Let DP and IP be the decomposition and inertia groups of P, respectively. Recall that σ 2 DP induces an elementσ ~ 2 G~, where G~ denotes the Galois group of OL=P over OK =P . Since P is unramified in L, it follows that jIPj = ePjP = 1, ∼ and then we have DP = G~ since IP is the kernel of the surjective homomorphism. It is clear that if OK =P has q elements, then G~ is a cyclic group with generator q being the Frobenius map x 7! x . Thus there is a unique σ 2 DP which maps to the Frobenius map. Since q = N(p) by definition, σ satisfies the desired condition N(P ) σ(α) ≡ α mod P 8α 2 OL: The uniqueness follows easily from the fact that any σ satisfying this condition must lie in DP. Definition 1.7. The unique element σ in lemma 1.6 is called the Artin symbol and is denoted by ((L=K)=P). The Artin symbol has the following useful properties: Corollary 1.8. Let K ⊂ L be a Galois extension, and let P be an unramified prime of K. Given a prime P of L containing P , we have: INTRODUCTION TO CLASS FIELD THEORY 3 (i) If σ 2 Gal(L=K), then L=K L=K = σ σ−1: σ(P) P (ii) The order of ((L=K)=P) is the inertial degree f = fPjP . (iii) P splits completely in L if and only if ((L=K)=P) = 1. Proof. For (i), notice that L=K σ σ−1(α) ≡ σ((σ−1(α))N(P)) ≡ αN(P) mod P P However, L=K (α) ≡ αN(σ(P)) mod P σ(P) and N(P) = N(σ(P)). Therefore, the uniqueness of Artin symbol shows that L=K L=K = σ σ−1: σ(P) P For (ii), since P is unramified, the decomposition group is isomorphic to the Galois group of OL=P over OK =P whose degree is the inertial degree f. Since the Artin symbol maps to a generator of the Galois group, it follows that the Artin symbol has order f. For (iii), notice that P splits completely in L if and only if e = f = 1. Since we have already assumed that e = 1, (iii) follows trivially from (ii). When K ⊂ L is an abelian extension, the Artin symbol ((L=K)=P) depends 0 only on the underlying prime P = P \OK . To see this, let P be another prime containing P . Notice that P0 = σ(P) for some σ 2 Gal(L=K). Then corollary 1.8 implies that L=K L=K L=K L=K = = σ σ−1 = P0 σ(P) P P since Gal(L=K) is abelian. It follows that whenever K ⊂ L is abelian, the Artin symbol can be written as ((L=K)=P ). As an example of the use of the Artin symbol, we can prove quadratic reciprocity without much effort: Lemma 1.9. The cyclotomic extension Q(ζn) is an abelian extension of Q. In ∼ ∗ fact, Gal(Q(ζn)=Q) = (Z=nZ) . Proof. This is a standard result on cyclotomic fields. For a proof, see [5]. Theorem 1.10. If p; q be distinct odd primes in Z, then p q (p−1)(q−1) = (−1) 4 : q p p Proof. Consider the cyclotomic extension Q ⊂ Q(ζp). If q 6= p, then x − 1 is separable modulo q since gcd(xp − 1; pxp−1) = 1. Therefore, q is unramified in p−1 Q(ζp). Moreover, 1; ζ; : : : ; ζ are distinct modulo q. But by the definition of the Artin symbol, ( (ζ )= ) Q p Q (ζ) ≡ ζN(q) ≡ ζq mod q: q 4 BOWEN WANG q Hence by the uniqueness of the Artin symbol, ((Q(ζp)=Q)=q) is the map ζ 7! ζ . p−1 Notice that (q=p) ≡ q 2 mod p by Euler's criterion. Therefore, q is a square p−1 ∗ mod p if and only if q is in the subgroup of order 2 of (Z=pZ) . But then the ∼ ∗ isomorphism Gal(Q(ζp)=Q) = (Z=ppZ) tells us that ((Q(ζp)=Q)=q) fixes the unique quadratic subextension K0 = Q( m) with m 2 Z squarefree. But since p is the 0 only prime that ramifies in Q(ζp), it follows that p ramifies in K , which means p−1 0 that m = (−1) 2 p. Therefore, ((K =Q)=q) = 1 and therefore q splits completely in K0. This implies that 2 p−1 x − (−1) 2 p ≡ 0 mod q has solution in Z, which means that p−1 ! (−1) 2 p q = : q p Now we define the Artin map in terms of the Artin symbol: Definition 1.11. Let m be a modulus divisible by all ramified primes of an abelian extension K ⊂ L. Given a prime p not dividing m, we have the Artin symbol L=K 2 Gal(L=K) p and it extends by multiplicativity to give a homomorphism Φm : IK (m) ! Gal(L=K) which is called the Artin map for K ⊂ L. When we want to refer explicitly to the extension involved, we will write ΦL=K;m instead of Φm. Now in order to see the correspondence between generalized ideal class groups and Galois groups of abelian extensions, we need the following three important theorems of class field theory. The proofs can be found in Janusz's book [4]. Theorem 1.12 (Artin Reciprocity Theorem). Let K ⊂ L be an abelian extension, and let m be a modulus divisible by all primes of K, finite or infinite, that ramifies in L.
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