### The GauSum and its Applications to Number Theory

#### 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 Gausums. 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 em>

_{r}

_{e}

_{G\{}

_{i}*(*

_{}}*x*

*-*

*x*)

*, where*

^{r}*G*and

*i*denote respectively the Galois group of

*F/Q*and the identity embedding of

*F.*

#### Keywords

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ISSN: 1927-5129