Generalized Kummer theory and its applications

Komatsu, Toru

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 127-138.

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.

Classification : 11R20, 12E10, 12G05
Mots clés : Generic polynomial, Kummer theory, Artin symbol

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