Galois module structure of \(p\)th-power classes of extensions of degree \(p\)
Let \(p\) be a prime number (possibly 2). Let \(F\) be a field with characteristic not \(p\) and containing a primitive \(p\)th root of unity. For all cyclic extensions \(K/F\) of degree \(p\), the authors determine the Galois module structure of the group \(J:= K^\times/K^{\times p}\) of \(p\)th power classes of \(K\) (where \(K^\times= K\setminus\{0\}\)). To know the Galois module structure of \(J\) is a fundamental question related to both the structure of the Galois cohomology of \(F\) and the structure of the maximal \(p\)-extension of \(F\). Just as in the case of local fields, the structure depends upon only the norm subgroup, its intersection with the subgroup of \(p\)th roots of unity, and the multiplicative subgroup of the base field. The proofs require only Hilbert's Theorem 90, basic Kummer theory and linear algebra over the group ring \(\mathbb F_p[\text{Gal}(K/F)]\).
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