# 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 14.1.5 is a canonical isomorphism . By Section 14.1.1, the group is cyclic with canonical generator . The 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 14.1.8.) Just as the primes and decomposition groups are all conjugate, the Frobenius elements over a given prime are conjugate.

Proposition 14.2.1   For each , we have In particular, the Frobenius elements lying over a given prime are all conjugate.

Proof. Fix . For any we have . Multiply by we see that , which proves the proposition. 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 . It extends to a map from the free abelian group on unramified primes to the group (the fractional ideals of ). 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 induces 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 (which has Galois group ) and the subgroups of . Giving an abelian extension of is exactly the same as giving an integer and a subgroup of . Even more importantly, the reciprocity map is simply . This is a nice generalization of quadratic reciprocity: for , the for a prime depends in a simple way on nothing but .

William Stein 2004-05-06