Division quaternion algebras over some cyclotomic fields

2023/03/04 by Diana Savin, Savin, Diana
Computer Science · Mathematics · #11R04 #11R18 #11R32 #11R52 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Primary:11S15 #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2303.02497

openalex publication_date 2023/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p₁, p₂ be two distinct prime integers, let n be a positive integer, n≥ 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(p₁, p₂) to be a division algebra over the nth cyclotomic field ℚ(ξ_n), when n∈\3,4,6,7,8,9,11,12\ and also we obtain a characterization for a quaternion algebra H(p₁, p₂) to be a division algebra over the nth cyclotomic field ℚ(ξ_n), when n∈\5,10\. In the 4th section we obtain a complete characterization for a quaternion algebra H_ℚ(ξ_n)(p₁, p₂) to be a division algebra, when n=l^k, with l a prime integer, l≡ 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_ℚ(ξ_l)(p₁, p₂) to be a division algebra, when l is a Fermat prime number.

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