The Gauß Sum and its Applications to Number Theory
| Autor: | Shin-ichi Katayama, Toru Nakahara, Nadia Khan, Hiroshi Sekiguchi |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Journal of Basic & Applied Sciences. 14:230-234 |
| ISSN: | 1927-5129 1814-8085 |
| DOI: | 10.6000/1927-5129.2018.14.35 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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