Journal of Prime Research in Mathematics

# 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

### 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.