Generalized Kummer theory and its applications
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.

DOI : https://doi.org/10.5802/ambp.259
Classification : 11R20,  12E10,  12G05
Mots clés : Generic polynomial, Kummer theory, Artin symbol
@article{AMBP_2009__16_1_127_0,
author = {Komatsu, Toru},
title = {Generalized Kummer theory and its applications},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {127--138},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {16},
number = {1},
year = {2009},
doi = {10.5802/ambp.259},
mrnumber = {2514533},
zbl = {1188.11054},
language = {en},
url = {www.numdam.org/item/AMBP_2009__16_1_127_0/}
}
Komatsu, Toru. Generalized Kummer theory and its applications. Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 127-138. doi : 10.5802/ambp.259. http://www.numdam.org/item/AMBP_2009__16_1_127_0/

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