Journal of the Mathematical Society of Japan

The simplest quartic fields with ideal class groups of exponents less than or equal to 2

Stéphane R. LOUBOUTIN

Abstract

The simplest quartic fields are the real cyclic quartic number fields defined by the irreducible quartic polynomials $x^{4}-mx^{3}-6x^{2}+mx+1$, where $m$ runs over the positive rational integers such that the odd part of $m^{2}+16$ is squarefree. We give an explicit lower bound for their class numbers which is much better than the previous known ones obtained by A. Lazarus. Then, using it, we determine the simplest quartic fields with ideal class groups of exponents $\leq 2$.

Article information

Source
J. Math. Soc. Japan, Volume 56, Number 3 (2004), 717-727.

Dates
First available in Project Euclid: 2 October 2007

https://projecteuclid.org/euclid.jmsj/1191334082

Digital Object Identifier
doi:10.2969/jmsj/1191334082

Mathematical Reviews number (MathSciNet)
MR2071669

Zentralblatt MATH identifier
1142.11365

Citation

R. LOUBOUTIN, Stéphane. The simplest quartic fields with ideal class groups of exponents less than or equal to 2. J. Math. Soc. Japan 56 (2004), no. 3, 717--727. doi:10.2969/jmsj/1191334082. https://projecteuclid.org/euclid.jmsj/1191334082