Math 254a: Class Field Theory: an Overview
نویسنده
چکیده
Class field theory relates abelian extensions of a given number field K to certain generalized ideal class groups ofK. The fundamental tool for doing this is Frobenius elements, and the Artin map obtained from them. Recall from lecture 24 that if L/K is an abelian extension (i.e., a Galois extension with abelian Galois group), and p is a prime of OK unramified in OL, then we defined Fr(p) ∈ Gal(L/K) to be the unique element acting as Frobenius on OL/q over OK/p for any q lying over p. In particular, it generates Dq/p, which has order eq/pfq/p, so we observe:
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