Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Krull-Schmidt decompositions of unit groups of number fields

Asuka Kumon, Donghyeok Lim

Abstract

We prove that the Krull-Schmidt decomposition of the Galois module of the $p$-adic completion of algebraic units is controlled by the primes that are ramified in the Galois extension and the $S$-ideal class group. We also compute explicit upper bounds for the number of possible Galois module structures of algebraic units when the Galois group is cyclic of order $p^{2}$ or $p^{3}$.

On Krull-Schmidt decompositions of unit groups of number fields

Abstract

We prove that the Krull-Schmidt decomposition of the Galois module of the -adic completion of algebraic units is controlled by the primes that are ramified in the Galois extension and the -ideal class group. We also compute explicit upper bounds for the number of possible Galois module structures of algebraic units when the Galois group is cyclic of order or .
Paper Structure (7 sections, 16 theorems, 48 equations)

This paper contains 7 sections, 16 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. The Krull-Schmidt decompositions of unit groups in general Galois extensions
  3. The Krull-Schmidt decompositions of unit groups in cyclic $p$-extensions
  4. The number of possible Galois structures of units in cyclic $p$-extensions
  5. The counting argument of Burns
  6. Some consequences of genus theory
  7. Proof of Theorem \ref{['2nd']}

Key Result

Theorem A

Let $p$ be an odd prime. Let $F/K$ be a Galois extension of number fields with $p \, | \, [F:K]$. Set $S= R_{F/K} \cup \Sigma_{K, p} \cup \Sigma_{K, \infty}$ and let $P$ be a Sylow $p$-subgroup of $G_{F/K}$. Then, we have

Theorems & Definitions (32)

  • Theorem A
  • Theorem B
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['Burnsordinary']}
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 22 more