[Paper Review] Cyclic algebras and construction of some Galois modules

Ján Mináč, Andrew Schultz+1 more|arXiv (Cornell University)|Oct 25, 2004

Rings, Modules, and Algebras5 references3 citations

TL;DR

This paper constructs cyclic extensions K/F of degree p^n with Galois group G, proving that for any prime p and integer i < n, there exists a field extension where the F_pG-module J = K^×/K^×^p contains an indecomposable direct summand of dimension p^i + 1. The result establishes the existence of such summands for all permissible parameters, resolving a realizability question in Galois module theory.

ABSTRACT

Let p be a prime and suppose that K/F is a cyclic extension of degree p n with group G. Let J be the FpG-module K × /K ×p of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p i + 1 for some 0 ≤ i &amp;lt; n. We show that for all primes p and all 0 ≤ i &amp;lt; n, there exists a field extension K/F with a summand of dimension p i + 1.

Motivation & Objective

• To determine whether indecomposable direct summands of dimension p^i + 1 can exist in the F_pG-module J = K^×/K^×^p for cyclic extensions K/F of degree p^n.
• To resolve the realizability question of whether such summands exist for all primes p and all i < n.
• To construct explicit field extensions K/F where J contains an indecomposable summand of dimension p^i + 1.

Proposed method

• Utilizes the structure of cyclic extensions K/F of degree p^n with Galois group G to analyze the module J = K^×/K^×^p over F_pG.
• Applies results from previous work on conditions for indecomposable summands in J, focusing on dimension constraints.
• Employs class field theory and cohomological techniques to construct extensions where specific module summands arise.
• Demonstrates that for any p and i < n, a suitable extension can be built such that J contains a summand of dimension p^i + 1.
• Uses the uniqueness result from prior work — at most one such summand exists — to ensure the construction is precise.

Experimental results

Research questions

• RQ1 For a given prime p and integer i < n, does there exist a cyclic extension K/F of degree p^n such that the F_pG-module J = K^×/K^×^p contains an indecomposable direct summand of dimension p^i + 1?
• RQ2 Can such summands be realized over number fields for all permissible p and i < n?
• RQ3 Is the dimension p^i + 1 the only possible dimension for such indecomposable summands in this setting?

Key findings

• For every prime p and every i with 0 ≤ i < n, there exists a cyclic extension K/F of degree p^n such that the F_pG-module J = K^×/K^×^p contains an indecomposable direct summand of dimension p^i + 1.
• The existence of such summands is realized for all combinations of p and i < n, confirming the realizability of all theoretically possible dimensions.
• The construction ensures that at most one such indecomposable summand exists, consistent with earlier results.
• The method provides a uniform construction mechanism across all primes p and all valid i < n, demonstrating broad applicability.