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

The Gauß Sum and its Applications to Number Theory|

Pages 230-234

Nadia Khan1, Shin-Ichi Katayama2, Toru Nakahara3 and Hiroshi Sekiguchi4

1National University of Computer & Emerging Sciences Lahore campus, Pakistan; 2University of Tokushima, Japan; 3Saga University, Japan; 4Daiichi Tekkou Co., 5 Chome-3 Tokaimachi, Tokai, Aichi Prefecture 476-0015, Japan

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 xx 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 ÕreG\{i} (xx)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.