is determined by . Equating these gives the original version of the Theorem. A further generalization is to cohomology with non-abelian coefficients: that if H is either the general or special linear group over L, including , then Another generalization is to a scheme X: where is the group of isomorphism classes of locally free sheaves of -modules of rank 1 for the Zariski topology, and is the sheaf defined by the affine line without the origin considered as a group under multiplication. There is yet another generalization to Milnor K-theory which plays a role in Voevodsky's proof of the Milnor conjecture. Proof Let be cyclic of degree and generate . Pick any of norm By clearing denominators, solving is the same as showing that has as an eigenvalue. We extend this to a map of -vector spaces via The primitive element theorem gives for some . Since has minimal polynomial we can identify via Here we wrote the second factor as a -polynomial in . Under this identification, our map becomes That is to say under this map is an eigenvector with eigenvalue iff has norm . References Chapter II of J.S. Milne, Class Field Theory, available at his website . External links Theorems in algebraic number theory