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Division quaternion algebras over some cyclotomic fields

Diana Savin

Abstract

Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $ξ_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{3,4,6,7,8,9,11,12\right\}$ and also we obtain a characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{5,10\right\}$. In the 4th section we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{n}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $n=l^{k},$ with $l$ a prime integer, $l\equiv 3$ (mod $4$) and $k$ a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{l}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $l$ is a Fermat prime number.

Division quaternion algebras over some cyclotomic fields

Abstract

Let be two distinct prime integers, let be a positive integer, and let be a primitive root of order of the unity. In this paper we obtain a complete characterization for a quaternion algebra to be a division algebra over the th cyclotomic field , when and also we obtain a characterization for a quaternion algebra to be a division algebra over the th cyclotomic field , when . In the 4th section we obtain a complete characterization for a quaternion algebra to be a division algebra, when with a prime integer, (mod ) and a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra to be a division algebra, when is a Fermat prime number.
Paper Structure (5 sections, 33 theorems, 24 equations)

This paper contains 5 sections, 33 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Division quaternion algebras over a cyclotomic field $\mathbb{Q}\left(\xi_{n}\right),$ with $n$$\in$$\left\{3,4,5,6,7,8,9,10,11,12\right\}$
  4. Division quaternion algebras over a cyclotomic field $\mathbb{Q}\left(\xi_{n}\right),$ when $n=l^{k},$ with $l$ is a prime integer $l\equiv 3$ (mod $4$)
  5. Division quaternion algebras over a cyclotomic field $\mathbb{Q}\left(\xi_{l}\right),$ when $l$ is a prime integer, $l=2^{k}+1$

Key Result

Theorem 2.1

(ireland). Let$d\neq 0,1$be a free squares integer.Let$\mathcal{O}_{K}$be the ring of integers of the quadratic field$K=\mathbb{Q}\left(\sqrt{d}\right)$and$\Delta_{K}$be the discriminant of$K.$Let$p$be an odd prime integer. Then, we have: i) $p$is ramified in$\mathcal{O}_{K},$if and only if$p|\Delta

Theorems & Definitions (48)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 38 more