Magma has some rudimentary functions to aid computations in Galois cohomology of number fields.
S: [RngOrdIdl] Default: false
Let K be a number field and M:K to K be an automorphism of K furthermore, denote by k the fixed field of M, thus M generates the automorphism group of the relative cyclic extension K/k. For some element a in K, such that NK/k(a) = 1, this function will find some element b such that a=b/M(b). If S is given it should contain a sequence of prime ideals such that there exists some b in the S-unit group over S.
ClassGroup: BoolElt Default: false
Ramification: BoolElt Default: false
Let k be a normal number field with (abstract) automorphism group G. For a set of prime ideals S of k, which is closed under the action of the subgroup U of G, a process is created that allows working with the cohomology of the multiplicative group of k - partially represented by a group of S-units. If ClassGroup is given, the set S is enlarged to support the current generators of the class group. If Ramification is present, then all ramified primes are also included in S.During the computations with this object the set S can be increased to allow the representation of a larger number of elements.
Sub: GrpPerm Default: false
SetVerbose("Cohomology", n): Maximum: 2
For a cohomology process C as created by SUnitCohomologyProcess and a 2-cocycle l:U x U to k given as a Magma-function, decide if l is split, ie. if there exists a 1-cochain m:U to k such that δ m = l for the cohomological coboundary map δ. If Sub is given it has to be a subgroup of the automorphism group of the number field underlying the cohomology process, otherwise the full automorphism group is used. This allows to restrict a cocycle easily.As a fixed cocycle l assumes only finitely many values, we can consider it as a cocycle with values in some suitable S-unit group. Similarly, it is exists, m also has values in some S'-unit group for a potentially larger set S'. This function first tries to "remove" ideals from the support of l, to make the set S as small as possible. Then the set is enlarged to make sure that m, if exists, can be found with values in the S'=S-unit group. Since the final problem now involves only finitely generated abelian groups, it can be solved by Magma's general cohomology machinery.
Sub: GrpPerm Default: false
Let U be a subgroup of the automorphism group G of some number field k, l:U x U to k * a 2-cocycle and I some ideal in k. If Sub is given, U is taken to be Sub, otherwise U := G. Assuming that each element l(u, v) has a valuation at all ideals in the U-orbit of I, ie. we have a unique decomposition of ideals l(u, v) = Jx(u, v) A(u, v) for integers x(u, v) and ideals A(u, v) coprime to J for all J in IU. Then we can use l to define a cocycle with values in IU which is a finitely generated group. This function determines if this cocycle splits, and if so, computes a 1-cochain with values in IU for some fixed ordering of IU. The cochain and IU are returned on success.