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For fields F of characteristic not p containing a primitive pth root of unity, we determine the Galois module structure of the group of pth-power classes of K for all cyclic extensions K/F of degree p. The foundation of the study of the maximal p-extensions of fields K containing a primitive pth root of unity is a group of the pth-power classes of the… (More)
Let K be a cyclic Galois extension of degree p over a field F containing a primitive pth root of unity. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these embedding problems we prove conditions on solv-ability, formulas for… (More)
One of the fundamental questions in current field theory, related to Grothen-dieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of… (More)
We define motivic Artin L-functions and show that they specialize to the usual Artin L-functions under the trace of Frobenius. In the last section we use our L-functions to prove a motivic analogue of the Chebotarev density theorem.
In our previous paper we describe the Galois module structures of pth-power class groups K × /K ×p , where K/F is a cyclic extension of degree p over a field F containing a primitive pth root of unity. Our description relies upon arithmetic invariants associated with K/F. Here we construct field extensions K/F with prescribed arithmetic invariants, thus… (More)
Here something stubborn comes, Dislodging the earth crumbs And making crusty rubble. It comes up bending double, And looks like a green staple. It could be seedling maple, Or artichoke, or bean. That remains to be seen. —Richard Wilbur, " Seed Leaves " (1–8) Whether reading or writing mathematics, it is over the simplest theorems that we linger. Something… (More)
Let E be a cyclic extension of pth-power degree of a field F of characteristic p. For all m, s ∈ N, we determine KmE/p s KmE as a (Z/p s Z)[Gal(E/F)]-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero.
We investigate the first two Galois cohomology groups of p-extensions over a base field which does not necessarily contain a primitive pth root of unity. We use twisted coefficients in a systematic way. We describe field extensions which are classified by certain residue classes modulo p n th powers of a related field, and we obtain transparent proofs and… (More)