## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 38, Number 2 (2015), 189-199.

### Refined version of Hasse's Satz 45 on class number parity

#### Abstract

For an imaginary abelian field $K$, Hasse [3, Satz 45] obtained a criterion for the relative class number to be odd in terms of the narrow class number of the maximal real subfield $K^+$ and the prime numbers which ramify in $K$, by using the analytic class number formula. In [4], we gave a refined version (= "$\Delta$-decomposed version") of Satz 45 by an algebraic method. In this paper, we give one more algebraic proof of the refined version.

#### Article information

**Source**

Tsukuba J. Math., Volume 38, Number 2 (2015), 189-199.

**Dates**

First available in Project Euclid: 15 April 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1429103720

**Digital Object Identifier**

doi:10.21099/tkbjm/1429103720

**Mathematical Reviews number (MathSciNet)**

MR3336267

**Zentralblatt MATH identifier**

1325.11111

**Subjects**

Primary: 11R18: Cyclotomic extensions

Secondary: 11R29: Class numbers, class groups, discriminants

**Keywords**

Class number parity abelian field reflection argument

#### Citation

Ichimura, Humio. Refined version of Hasse's Satz 45 on class number parity. Tsukuba J. Math. 38 (2015), no. 2, 189--199. doi:10.21099/tkbjm/1429103720. https://projecteuclid.org/euclid.tkbjm/1429103720