[Paper Review] Galois Module Structure of $p$th-Power Classes of Extensions of Degree p

Ján Mináč, John Swallow|ArXiv.org|Feb 21, 2002

Rings, Modules, and Algebras4 citations

TL;DR

This paper determines the Galois module structure of the group of $p$th-power classes $J = K^\times / K^{\times p}$ for cyclic extensions $K/F$ of degree $p$, where $F$ is a field of characteristic not $p$ containing a primitive $p$th root of unity. Using Kummer theory, Hilbert's Theorem 90, and linear algebra over $\mathbb{F}_p[\operatorname{Gal}(K/F)]$, it shows that $J$ decomposes uniquely as a direct sum of indecomposable modules determined by the norm subgroup $N(K^\times)$, the presence of $\xi_p$ in $N(K^\times)$, and the structure of $F^\times / N(K^\times)$, providing a complete classification for both odd $p$ and $p=2$. The key contribution is a complete, explicit, and elementary description of the Galois module structure of $J$ in terms of invariants of the base field and the extension.

ABSTRACT

For fields F of characteristic not p containing a primitive $p$th root of unity, we determine the Galois module structure of the group of $p$th-power classes of K for all cyclic extensions K/F of degree p.

Motivation & Objective

• To determine the Galois module structure of the group $J = K^\times / K^{\times p}$ for all cyclic extensions $K/F$ of degree $p$, where $F$ contains a primitive $p$th root of unity.
• To provide a complete and explicit classification of the $\mathbb{F}_p[\operatorname{Gal}(K/F)]$-module structure of $J$ in terms of invariants of the base field and the extension.
• To show that this structure depends only on the norm subgroup $N(K^\times)$, the presence of $\xi_p$ in $N(K^\times)$, and the quotient $F^\times / N(K^\times)$, using only elementary tools.
• To establish that the decomposition of $J$ into indecomposable $\mathbb{F}_p[G]$-modules is unique, leveraging the Krull-Schmidt theorem.

Proposed method

• Use of Kummer theory to identify $J = K^\times / K^{\times p}$ with the Galois cohomology group $H^1(G, \mu_p)$, where $G = \operatorname{Gal}(K/F)$.
• Application of Hilbert's Theorem 90 to analyze the norm map $N: K^\times \to F^\times$ and its image $N(K^\times)$.
• Reduction of the Galois module structure of $J$ to linear algebra over the group ring $\mathbb{F}_p[G]$, using the action of $G = \langle \sigma \rangle$ of order $p$.
• Decomposition of $J$ into three parts: $X$ (indecomposable of dimension 1 or 2 depending on $\xi_p \in N(K^\times)$), $Y$ (free $\mathbb{F}_p[G]$-module with $Y^G = N(J)$), and $Z$ (trivial module).
• Use of the norm map $N$ as an element of $\mathbb{F}_p[G]$ and as a $G$-homomorphism to analyze the structure of $J$ and its fixed submodule $J^G$.

Experimental results

Research questions

• RQ1 What is the complete Galois module structure of $J = K^\times / K^{\times p}$ for a cyclic extension $K/F$ of degree $p$ when $F$ contains a primitive $p$th root of unity?
• RQ2 How does the structure of $J$ as an $\mathbb{F}_p[G]$-module depend on the norm subgroup $N(K^\times)$ and the presence of $\xi_p$ in $N(K^\times)$?
• RQ3 Can the decomposition of $J$ into indecomposable $\mathbb{F}_p[G]$-modules be made explicit and unique using only elementary tools from Kummer theory and linear algebra?
• RQ4 What conditions on $F^\times / N(K^\times)$ and $N(K^\times)$ determine whether $J$ contains free or trivial summands?

Key findings

• For $p > 2$, the $\mathbb{F}_p[G]$-module $J$ decomposes as $X \oplus Y \oplus Z$, where $X$ has dimension 1 if $\xi_p \in N(K^\times)$, and dimension 2 otherwise; $Y$ is a free module with $Y^G = N(J)$; and $Z$ is trivial.
• For $p = 2$, $X$ is 1-dimensional if $-1 \in N(K^\times)$, and trivial otherwise; $Y$ is free with $Y^G = N(J)$; $Z$ is trivial.
• The rank of any maximal free $\mathbb{F}_p[G]$-submodule of $J$ equals the $\mathbb{F}_p$-dimension of $N(J)$, which is $\dim_{\mathbb{F}_p}(F^\times / N(K^\times)) + 2\Upsilon(K)$ for $p > 2$, and $\dim_{\mathbb{F}_2}(F^\times / N(K^\times)) + \Upsilon(K)$ for $p = 2$, where $\Upsilon(K)$ counts certain $p$-rigid elements.
• The decomposition of $J$ into indecomposable $\mathbb{F}_p[G]$-modules is unique, as each cyclic summand is indecomposable and its endomorphism ring is local, satisfying the Krull-Schmidt theorem.
• The module $J$ is free if and only if $N(J)$ is trivial and $\xi_p \in N(K^\times)$ for $p > 2$, or $-1 \notin N(K^\times)$ for $p = 2$, with additional conditions on the structure of $F^\times / N(K^\times)$.
• Two such modules $J_1$ and $J_2$ for extensions $K_1/F_1$ and $K_2/F_2$ are isomorphic as $\mathbb{F}_p[G]$-modules if and only if $\dim_{\mathbb{F}_p}(F_1^\times / N(K_1^\times)) = \dim_{\mathbb{F}_p}(F_2^\times / N(K_2^\times))$ and $|\mathfrak{K}_2|$ is equal for both, where $\mathfrak{K}_2$ indexes the $p$-rigid elements in $F^\times / F^{\times p}$.