Nagoya Mathematical Journal

Values of zeta functions and class number 1 criterion for the simplest cubic fields

Hyung Ju Hwang and Hyun Kwang Kim

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Abstract

Let $K$ be the simplest cubic field defined by the irreducible polynomial $$f(x) = x^{3}+mx^{2}-(m+3)x+1,$$ where $m$ is a nonnegative rational integer such that $m^{2}+3m+9$ is square-free. We estimate the value of the Dedekind zeta function $\zeta_{K}(s)$ at $s = -1$ and get class number $1$ criterion for the simplest cubic fields.

Article information

Source
Nagoya Math. J., Volume 160 (2000), 161-180.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631504

Mathematical Reviews number (MathSciNet)
MR1804143

Zentralblatt MATH identifier
0999.11067

Subjects
Primary: 11R16: Cubic and quartic extensions
Secondary: 11R29: Class numbers, class groups, discriminants 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Citation

Kim, Hyun Kwang; Hwang, Hyung Ju. Values of zeta functions and class number 1 criterion for the simplest cubic fields. Nagoya Math. J. 160 (2000), 161--180. https://projecteuclid.org/euclid.nmj/1114631504


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