More explicitly, if , with a root of the irreducible polynomial , then the prime ideals correspond to the irreducible factors of in . The fields then correspond to adjoing roots of each of these irreducible factors of in . Note that for most , a generalization of Hensel's lemma (see Section 1.5.1) asserts that we can factor by factoring modulo and iteratively lifting the factorization.
We have a natural map got by restriction; implicit in this is a choice of embedding of in that sends into . We may thus view as a subgroup of .
Let be any
module. Then
this restriction map induces a restriction map on Galois cohomology
Likewise there is a restriction map for each
real Archimedian prime , i.e., for each embedding
we have a map
William 2007-05-25