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Hopf-Galois module structure of degree p extensions of p-adic fields

Daniel Gil-Muñoz

Abstract

Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition that the ring of integers $\mathcal{O}_L$ is free as a module over its associated order in the unique Hopf-Galois structure on $L/K$.

Hopf-Galois module structure of degree p extensions of p-adic fields

Abstract

Let be an odd prime number. For a degree extension of -adic fields , we give a complete characterization of the condition that the ring of integers is free as a module over its associated order in the unique Hopf-Galois structure on .

Paper Structure

This paper contains 23 sections, 37 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hopf-Galois extensions
  4. Extensions of $p$-adic fields and their module structure
  5. Scaffolds on a degree $p$ extension
  6. Continued fractions and distance to the nearest integer
  7. Reduction to the totally ramified case
  8. The maximally ramified case
  9. A method to find the associated order
  10. $H$-Kummer extensions
  11. The maximality of $\mathfrak{A}_{L/K}$
  12. Typical degree $p$ extensions
  13. The only Hopf-Galois structure
  14. The underlying Hopf algebra
  15. The action on the extension
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $L/K$ be a totally ramified cyclic degree $p$ extension of $p$-adic fields and let $e$ be the ramification index of $K/\mathbb{Q}_p$. Let $G\coloneqq\mathrm{Gal}(L/K)=\langle\sigma\rangle$, let $t$ be the ramification jump of $L/K$ and let $a$ be the remainder of $t$ mod $p$.

Figures (3)

  • Figure 1: Point $M_h$ assigned to $h\in\mathbb{Z}$.
  • Figure 2: The distance between $M_h$ and $M_k$.
  • Figure 3: Point $M_h$ between $M_{h'}$ and $M_{h"}$.

Theorems & Definitions (67)

  • Theorem 1.1: F. Bertrandias, J.P. Bertrandias, M.J. Ferton
  • Theorem 1.2
  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 57 more