The Gauß Sum and its Applications to Number Theory - Pages 230-234

Nadia Khan, Shin-Ichi Katayama, Toru Nakahara and Hiroshi Sekiguchi

https://doi.org/10.6000/1927-5129.2018.14.35

Published: 22 June 2018

Abstract: The purpose of this article is to determine the monogenity of families of certain biquadratic fields K and cyclic bicubic fields L obtained by composition of the quadratic field of conductor 5 and the simplest cubic fields over the field Q of rational numbers applying cubic Gauß sums. The monogenic biquartic fields K are constructed without using the integral bases. It is found that all the bicubic fields L over the simplest cubic fields are non-monogenic except for the conductors 7 and 9. Each of the proof is obtained by the evaluation of the partial differents x-x^r of the different ¶_F/Q (x) with F=K or L of a candidate number x, which will or would generate a power integral basis of the fields F. Here r denotes a suitable Galois action of the abelian extensions F/Q and ¶_F/Q (x) is defined by Õ_reG\{i} (x-x)^r, where G and i denote respectively the Galois group of F/Q  and the identity embedding of F.

Keywords: Monogenity, Biquadratic field, Simplest cubic field, Cyclic sextic field, Discriminant, Integral basis.