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Non-Abelian extensions of degree $p^3$ and $p^4$ in characteristic $p>2$

Grant Moles

Abstract

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order $p^3$ or the ten non-Abelian groups of order $p^4$, $p$ an odd prime, over a field of characteristic $p$.

Non-Abelian extensions of degree $p^3$ and $p^4$ in characteristic $p>2$

Abstract

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order or the ten non-Abelian groups of order , an odd prime, over a field of characteristic .
Paper Structure (3 sections, 6 theorems, 47 equations)

This paper contains 3 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Extensions of degree $p^3$
  3. Extensions of degree $p^4$

Key Result

Theorem 1.1

Let $K_4/K_0$ be a non-Abelian Galois extension of degree $p^4$ with $\sigma_i$, $K_i$, and $x_i$ for $1\leq i\leq 4$ as defined above. Then $x_1$ and $x_2$ can be chosen so that $\wp(x_1)\in K_0$ and $\wp(x_2)\in K_0$, where $\wp$ is the Weierstrass $\wp$ function, defined by $\wp(x)=x^p-x$. Denote for some $\beta_i\in K_0$, where $D_1\in\mathbb F_p[x]$ the Witt polynomial For each group, the sp

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 1 more