File:ClassExtensionAndArtinTransfer.png

English: This is the famous diagram whose commutativity was proved by Emil Artin. It reduces the number theoretic problem of the principalization of ideal classes of a base field K in an extension field L to the purely group theoretic task to determine the kernel of the transfer homomorphism T from the Galois group G=Gal(K'/K) to the Galois group H=Gal(L'/L), where K', resp. L', denote maximal abelian extensions of K, resp. L. The horizontal arrows denote the reciprocity isomorphisms between class groups and Galois groups, mapping the class of a prime ideal p to the Frobenius automophism of p. The map j is the class extension homomorphism from K to L.