# Frobenius Elements

Suppose that is a finite Galois extension with group and is a prime such that (i.e., an unramified prime). Then for any , so the map of Theorem 9.3.5 is a canonical isomorphism . By Section 9.3.1, the group is cyclic with canonical generator . The Frobenius element corresponding to is . It is the unique element of such that for all we have (To see this argue as in the proof of Proposition 9.3.8.) Just as the primes and decomposition groups are all conjugate, the Frobenius elements corresponding to primes are all conjugate as elements of .

Proposition 9.4.1   For each , we have In particular, the Frobenius elements lying over a given prime are all conjugate.

Proof. Fix . For any we have . Applying to both sides, we see that , so . Thus the conjugacy class of in is a well-defined function of . For example, if is abelian, then does not depend on the choice of lying over and we obtain a well defined symbol called the Artin symbol. It extends to a homomorphism from the free abelian group on unramified primes to . Class field theory (for ) sets up a natural bijection between abelian Galois extensions of and certain maps from certain subgroups of the group of fractional ideals for . We have just described one direction of this bijection, which associates to an abelian extension the Artin symbol (which is a homomorphism). The Kronecker-Weber theorem asserts that the abelian extensions of are exactly the subfields of the fields , as varies over all positive integers. By Galois theory there is a correspondence between the subfields of the field , which has Galois group , and the subgroups of , so giving an abelian extension of is exactly the same as giving an integer and a subgroup of . The Artin reciprocity map is then .

William Stein 2012-09-24