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Cyclotomic function fields over finite fields with irreducible quadratic modulus

Nazar Arakelian, Luciane Quoos

Abstract

Let $\mathbb{F}_q$ be the finite field of order $q$ and $F=\mathbb{F}_q(x)$ the rational function field. In this paper, we give a characterization of the cyclotomic function fields $F(Λ_M)$ with modulus $M$, where $M \in \mathbb{F}_q[T]$ is a monic and irreducible polynomial of degree two. We also provide the full automorphism group of $F(Λ_M)$ in odd characteristic, extending results of \cite{MXY2016} where the automorphism group of $F(Λ_M)$ over $\mathbb{F}_q$ was computed.

Cyclotomic function fields over finite fields with irreducible quadratic modulus

Abstract

Let be the finite field of order and the rational function field. In this paper, we give a characterization of the cyclotomic function fields with modulus , where is a monic and irreducible polynomial of degree two. We also provide the full automorphism group of in odd characteristic, extending results of \cite{MXY2016} where the automorphism group of over was computed.
Paper Structure (6 sections, 14 theorems, 57 equations)

This paper contains 6 sections, 14 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Automorphims of function fields
  4. Cyclotomic function fields
  5. The characterization of $F(\Lambda_M)$
  6. The full automorphism group

Key Result

Theorem 2.1

HKT Let $\mathcal{F}$ be an algebraic function field over $\overline{\mathbb{F}}_q$ and $G_P \leq \operatorname{Aut}_{\overline{\mathbb{F}}_q}(\mathcal{F})$ be the stabilizer of a place $P$ of $\mathcal{F}$. Then the $p$-Sylow subgroup $S_p$ of $G_P$ is a normal subgroup and the quotient group $G_P/

Theorems & Definitions (28)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 18 more