The Gauß Sum and its Applications to Number Theory - Nadia Khan; Shin-Ichi Katayama; Toru Nakahara; Hiroshi Sekiguchi (Journal of Basic & Applied Sciences, 2018)
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 !"!# of the different !F/Q(") with F=K or L of a candidate number !, which will or would generate a power integral basis of the fields F.Here ! denotes a suitable Galois action of the abelian extensions F/Q and !F/Q(") is defined by !"G{#(%&%!),where G and ! denote respectively the Galois group of F/Q and the identity embedding of F.