Journal of Prime Research in Mathematics
Vol. 18 (2022), Issue 2, pp. 112 – 124
ISSN: 1817-3462E (Online) 1818-5495 (Print)
ISSN: 1817-3462E (Online) 1818-5495 (Print)
Algebraic integers of pure sextic extensions
Antonio Aparecido de Andrade\(^{1,*}\), Linara St´efani Facini\(^1\),Livea Cichito Esteves\(^1\)
\(^a\) Department of Mathematics, S˜ao Paulo State University (Unesp), Institute of Biosciences, Humanites and Exact Sciences (Ibilce),Campus S˜ao Jos´e do Rio Preto, S˜ao Paulo, Brazil
Correspondence should be addressed to: : antonio.andrade@unesp.br
Copyright © 2022 Antonio Aparecido de Andrade, Linara St´efani Facini,Livea Cichito Esteves. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Published: Received: 24 October 2022; Accepted: 03 December 2022; Published Online: 21 December 2022.
Abstract
Let K = Q(θ), where θ = √6 d, be a pure sextic field with d ̸= 1 a square free integer. In this paper, we characterize completely whether {1, θ, . . . , θ5} is an integral basis of K or do not. When d ̸≡ ±1, ±17, ±10, −15, −11, −7, −3, 5, 13(mod 36) we prove that K has a power integral basis. Furthermore, for the other cases we present an integral basis.
Keywords:
Algebraic number field, algebraic number integer, pure sextic extension.