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The Gauß Sum and its Applications to Number Theory.
Author(s):
1. Nadia Khan: National University of Computer & Emerging Sciences Lahore campus, Pakistan
2. Shin-Ichi Katayama: University of Tokushima, Japan
3. Toru Nakahara: Saga University, Japan
4. Hiroshi Sekiguchi: Daiichi 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 theintegral bases. It is found that all the bicubic fields L over the simplest cubic fields are non-monogenic except for theconductors 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.
Page(s): 230-234
DOI: DOI not available
Published: Journal: Journal of Basic and Applied Sciences, Volume: 14, Issue: 0, Year: 2018
Keywords:
NULLL
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