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A family of infinite pairs of quadratic fields ℚ(√D) and ℚ(√-D) whose class numbers are both divisible by 3

Toru Komatsu — 2001

Acta Arithmetica

An infinite family of pairs of quadratic fields ℚ(√D) and ℚ(√mD) whose class numbers are both divisible by 3

Toru Komatsu — 2002

Acta Arithmetica

Generalized Kummer theory and its applications

Toru Komatsu — 2009

Annales mathématiques Blaise Pascal

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 $ζ \notin k$ and $ω \in k$ where $ζ$ is a primitive $n$ -th root of unity and $ω = ζ + ζ^{- 1}$ . In particular, this result with $ζ \in k$ 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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