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Totally ramified subfields of $p$-Algebras

Adam Chapman, S. Srimathy

Abstract

We conjecture that a $p$-algebra over a complete discrete valued field $K$ contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyclic maximal subfield. We prove the conjecture in several cases.

Totally ramified subfields of $p$-Algebras

Abstract

We conjecture that a -algebra over a complete discrete valued field contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyclic maximal subfield. We prove the conjecture in several cases.
Paper Structure (12 sections, 14 theorems, 22 equations)

This paper contains 12 sections, 14 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Cyclic extension of degree $p^n$ over fields of characteristic $p$
  5. Symbol manipulation in cyclic p-algebras
  6. Ramification of Division algebras over complete discrete valued fields
  7. From cyclic to purely inseparable
  8. From purely inseparable to cyclic
  9. When $k$ is perfect
  10. When the degree of $A$ is $p$ or $p^2$
  11. When $\dim_{\mathbb{F}_p}(k/\mathcal{P}(k)) \geq 2$
  12. Existence of totally ramified separable maximal subfields

Key Result

Lemma 3.1

Let $F$ be a field of characteristic $p$. Suppose that $\dim_{\mathbb{F}_p}(F/\mathcal{P}(F)) \geq 2$. Then for any $m\geq 0$, there exists cyclic extensions $L_1/F$ and $L_2/F$ of degree $p^m$ satisfying $L_1 \cap L_2 = F$

Theorems & Definitions (27)

  • Conjecture 1.1
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • Remark 5.2
  • ...and 17 more