Math 254a: Class Field Theory Iii

1. The main theorems revisited We now state in more detail the main theorems of class field theory. However, we need two more preliminary definitions: Definition 1.1. Given an extension L/K of number fields, and a fractional ideal I of L, we define the fractional ideal N L/K (I) of K as follows: write I = i q ei i , let p i = q i ∩ O K , and let N L/K (I) = i p eifi i. Definition 1.2. Let σ : K → R be a real imbedding. We say that σ is ramified in an extension L if σ extends to a complex imbedding of L. We now state the main theorems of class field theory with more precision than before. We emphasize that although the statements are simple, the proofs are quite deep, and would require the better part of a semester-long course to cover in full. Theorem 1.3. (Artin reciprocity) Let L/K be an abelian extension of number fields, and S the set of primes of K unramified in L (including real imbeddings). Then there exists a modulus m such that m 1 = S ∩ M 0 K , m ∞ = M ∞ K and the kernel of the Artin map Art : I K (m) → Gal(L/K) is a congruence subgroup for m. More precisely, the kernel is equal to the subgroup of I K (m) generated by P K,1 (m) and by N L/K (I) as I ranges over fractional ideals of L which are prime to m 0. Finally, we may set m to be the conductor of ker Art, which is the smallest modulus for which ker Art is a congruence subgroup, and which we call also the conductor of L/K. Theorem 1.4. (Existence theorem) Let K be a number field, and m a modulus in K. For any congruence subgroup H for m, there exists a unique number field L which is an abelian extension of K, which has conductor dividing m (in particular, it is ramified only at primes contained in m 1 ∪ m ∞), and for which the Artin map I K (m) → Gal(L/K) has kernel exactly H. In fact, the conductor of L is equal to the conductor of H. Definition 1.5. Given any modulus m for K, ray class field K m is the abelian extension obtained from I …