## Journal of the Mathematical Society of Japan

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

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

#### Abstract

The simplest quartic fields are the real cyclic quartic number fields defined by the irreducible quartic polynomials ${x}^{4}-m{x}^{3}-6{x}^{2}+mx+1$, 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 $\le 2$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 2 October 2007

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/1191334082

**Mathematical Reviews number (MathSciNet)**

MR2071669

**Zentralblatt MATH identifier**

1142.11365

**Subjects**

Primary: 11R16: Cubic and quartic extensions 11R29: Class numbers, class groups, discriminants 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11Y40: Algebraic number theory computations

**Keywords**

quartic field simplest quartic field class number class group zeta function

#### 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