On the parity of the class number of a biquadratic field. (English) Zbl 0852.11065

Kučera, Radan

J. Number Theory 52, No. 1, 43-52 (1995).

The parity of the class number of any biquadratic field is characterized by P. E. Conner and J. Hurrelbrink [Class number parity, Ser. Pure Math. 8 (World Scientific, Singapore) (1988; Zbl 0743.11061)] up to the cases:
1) \(\mathbb{Q} (\sqrt {p}, \sqrt {q})\), where \(p\) and \(q\) are different primes \(p\equiv q\equiv 1\pmod 4\), the Legendre symbol \((p/q )=1\);
2) \(\mathbb{Q} (\sqrt {p}, \sqrt {2})\), where \(p\) is a prime, \(p\equiv 1\pmod 8\).
The problem of characterizing fields with even class number among these fields is equivalent to the problem of characterizing fields \(\mathbb{Q} (\sqrt {pq})\) (case 1) and \(\mathbb{Q} (\sqrt {2p})\) (case 2) with a class number divisible by 4. The author gives solutions for these problems.

Reviewer: T.Lepistö (Tampere)

MSC:

11R29	Class numbers, class groups, discriminants
11R16	Cubic and quartic extensions

Keywords:

parity of the class number; biquadratic field

Citations:

Zbl 0743.11061