On the parity of the class number of the field \({\mathbb Q}(\sqrt p,\sqrt q,\sqrt r)\). (English) Zbl 1024.11071

Bulant, Michal

Acta Math. Inform. Univ. Ostrav. 6, No. 1, 41-52 (1998).

This article extends the author’s earlier work [J. Number Theory 68, 72-86 (1998; Zbl 0913.11046)] where \(p,q,r\) were all congruent to \(1\bmod 4\). The main result of the paper under review gives necessary and sufficient conditions for the class number of the field \({\mathbb Q}(\sqrt p,\sqrt q,\sqrt r)\) to be even. Here \(p\), \(q\), and \(r\) are different primes either congruent to \(1\) modulo \(4\) or equal to \(2\). These conditions depend on the combinations of the values of the Kronecker symbol \((a/b)\) for \(a,b\in \{p,q,r\}\) and on the Dirichlet characters modulo \(l\) (\(l\in \{p,q,r\}\); modulo 16 if \(l=2\)) of order 4.
The method of R. Kučera [J. Number Theory 52, 43-52 (1995; Zbl 0852.11065)] is used.

Reviewer: Ladislav Skula (Brno)

MSC:

11R29	Class numbers, class groups, discriminants
11R20	Other abelian and metabelian extensions

Keywords:

class number; cyclotomic unit; crossed homomorphism

Citations:

Zbl 0913.11046; Zbl 0852.11065

References:

[1]	M. Bulant. : On the parity of the class number of the field \(\mathbb Q(\sqrt{p}, \sqrt{q}, \sqrt{r})\). J. Number Theory, 68(1):72-86, Jan. 1998. · Zbl 0913.11046
[2]	R. Kučera. : On the parity of the class number of a biquadratic field. J. Number Theory, 52(1):43-52, May 1995. · Zbl 0852.11065
[3]	R. Kučera. : On the Stickelberger ideal and circular units of a compositum of quadratic fields. J. Number Theory, 56(1):139-166, Jan. 1996. · Zbl 0840.11044