The Gauß Sum and its Applications to Number Theory

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 Gaus 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 O r e G\{ i } ( x - x ) r , where G and i denote respectively the Galois group of F/Q  and the identity embedding of F.