[Paper Review] Construction and classification of some Galois modules

Ján Mináč, John Swallow|ArXiv.org|Apr 15, 2003

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper completes the classification of Galois module structures on $K^\times/K^{\times p}$ for cyclic extensions $K/F$ of degree $p$ over fields containing a primitive $p$th root of unity. By constructing field extensions with prescribed arithmetic invariants—$d = \dim_{\mathbb{F}_p} F^\times / N(K^\times)$, $e = \dim_{\mathbb{F}_p} N(K^\times)/F^{\times p}$, and $\Upsilon = 1$ iff $\xi_p \in N(K^\times)$—the authors establish a complete characterization of all possible $\mathbb{F}_p[\operatorname{Gal}(K/F)]$-module structures on $K^\times/K^{\times p}$, proving realizability conditions for these invariants.

ABSTRACT

In our previous paper we describe the Galois module structures of $p$th-power class groups $K^ imes/{K^{ imes p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th 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 completing our classification of Galois modules $K^{ imes}/K^{ imes p}$.

Motivation & Objective

• To complete the classification of $\mathbb{F}_p[\operatorname{Gal}(K/F)]$-module structures on $K^\times/K^{\times p}$ for cyclic extensions $K/F$ of degree $p$.
• To construct explicit field extensions $K/F$ with given values of the arithmetic invariants $d$, $e$, and $\Upsilon$.
• To determine the necessary and sufficient conditions on $d$, $e$, and $\Upsilon$ for such extensions to exist, thereby realizing all possible module structures.

Proposed method

• Use of Galois module theory and cohomological techniques to analyze the structure of $K^\times/K^{\times p}$ as an $\mathbb{F}_p[\operatorname{Gal}(K/F)]$-module.
• Definition and analysis of three arithmetic invariants: $d = \dim_{\mathbb{F}_p} F^\times / N(K^\times)$, $e = \dim_{\mathbb{F}_p} N(K^\times)/F^{\times p}$, and $\Upsilon = 1$ iff $\xi_p \in N(K^\times)$.
• Construction of field extensions via inverse limit and embedding techniques, particularly using $\mathbb{Z}/p^2\mathbb{Z}$-extensions to control the norm condition for $\Upsilon$.
• Application of Galois cohomology and the Kummer isomorphism to relate the module structure to the invariants.
• Use of the norm map $N: K^\times \to F^\times$ and its image to define and compute $d$ and $e$.
• Leveraging results from prior work ([MS], Theorem 3, Corollary 2) to establish isomorphism types of modules in terms of direct sums of cyclic modules $M_{i,j}$.

Experimental results

Research questions

• RQ1 Which triples $(d, e, \Upsilon)$ of arithmetic invariants can be realized by a cyclic extension $K/F$ of degree $p$ with $F$ containing a primitive $p$th root of unity?
• RQ2 How do the invariants $d$, $e$, and $\Upsilon$ determine the isomorphism class of the $\mathbb{F}_p[\operatorname{Gal}(K/F)]$-module $K^\times/K^{\times p}$?
• RQ3 What conditions on $d$, $e$, and $\Upsilon$ are necessary and sufficient for the existence of such a cyclic extension $K/F$?
• RQ4 How do the module structures differ for $p=2$ versus $p>2$, particularly in terms of uniqueness of invariants?
• RQ5 Can every possible $\mathbb{F}_p[G]$-module structure on $K^\times/K^{\times p}$ be realized via a suitable cyclic extension $K/F$?

Key findings

• For any prime $p$, and for any cardinal numbers $d$, $e$, and $\Upsilon \in \{0,1\}$, there exists a cyclic extension $K/F$ of degree $p$ realizing the invariants $d$, $e$, and $\Upsilon$ if and only if: $\Upsilon = 0$ implies $d \geq 1$, $p > 2$ implies $e \geq 1$, and if $p = 2$ and $\Upsilon = 1$, then $e \geq 1$.
• For $p > 2$, the $\mathbb{F}_p[G]$-module $J = K^\times/K^{\times p}$ is completely determined up to isomorphism by the invariants $d$, $e$, and $\Upsilon$, and is isomorphic to $\left(\bigoplus_{j \in \mathfrak{K}_1} M_{1,j}\right) \oplus \left(\bigoplus_{j \in \mathfrak{K}_2} M_{2,j}\right) \oplus \left(\bigoplus_{j \in \mathfrak{K}_p} M_{p,j}\right)$ with $|\mathfrak{K}_1| + 1 = 2\Upsilon + d$, $|\mathfrak{K}_2| = 1 - \Upsilon$, and $|\mathfrak{K}_p| + 1 = e$.
• For $p = 2$, the module $J$ is isomorphic to $\left(\bigoplus_{j \in \mathfrak{K}_1} M_{1,j}\right) \oplus \left(\bigoplus_{j \in \mathfrak{K}_2} M_{2,j}\right)$ with $|\mathfrak{K}_1| + 1 = 2\Upsilon + d$ and $|\mathfrak{K}_2| + \Upsilon = e$, and is uniquely determined by $2\Upsilon + d$ and $e - \Upsilon$ (or $e$ if infinite).
• The invariants $d$, $e$, and $\Upsilon$ are independent of the choice of isomorphism $G \cong \operatorname{Gal}(K/F)$, ensuring the module structure is well-defined.
• The construction of such extensions is possible over fields of finite transcendence degree over $\mathbb{Q}$, with $\operatorname{tr.deg}(F/\mathbb{Q}) \leq 1 + \max\{e, d+1\}$, when $d, e$ are finite.
• A counterexample shows that for $p = 2$, isomorphic modules may have different arithmetic invariants: $K_1^\times/K_1^{\times 2} \cong \mathbb{F}_2$ as $\mathbb{F}_2[G]$-modules, but with invariants $(d,e,\Upsilon) = (0,1,1)$ and $(2,0,0)$ respectively, illustrating non-uniqueness of invariants under isomorphism in the $p=2$ case.