A relative version of Kummer theory

TL;DR

This paper extends Kummer theory by demonstrating that the Galois module structures involved in Kummer duality are preserved in cyclic Galois extensions of degree a power of a prime, emphasizing a module-theoretic perspective.

Contribution

It establishes a relative version of Kummer theory showing Galois module structures are preserved under Kummer duality in cyclic extensions of prime power degree.

Findings

Galois module structures are isomorphic on both sides of the Kummer pairing.

Kummer duality holds at the level of finitely generated Galois modules.

The result applies to cyclic Galois extensions of degree p^l.

Abstract

Let $E/F$ be a cyclic Galois extension of degree $p^l$ with Galois group $G$. It is shown that the Galois module structure of both sides of the Kummer pairing (for Kummer extensions of $E$) are the same. In other words, we show that the Kummer duality holds in the level of finitely generated $G$-modules.