The Gauß Sum and its Applications to Number Theory


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

How to Cite

Nadia Khan, Shin-Ichi Katayama, Toru Nakahara, & Hiroshi Sekiguchi. (2018). The Gauß Sum and its Applications to Number Theory . Journal of Basic & Applied Sciences, 14, 230–234.


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.


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