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Blaise Pascal
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Toru Komatsu
Generalized Kummer theory and its applications
Annales mathématiques Blaise Pascal, 16 no. 1 (2009), p. 127-138, doi: 10.5802/ambp.259
Article PDF | Reviews MR 2514533 | Zbl 1188.11054
Class. Math.: 11R20, 12E10, 12G05
Keywords: Generic polynomial, Kummer theory, Artin symbol

Résumé - Abstract

In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order $n$ and obtain a generalized Kummer theory. It is useful under the condition that $\zeta \notin k$ and $\omega \in k$ where $\zeta $ is a primitive $n$-th root of unity and $\omega =\zeta +\zeta ^{-1}$. In particular, this result with $\zeta \in k$ implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.

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