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

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.

### MSC:

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

### Keywords:

parity of the class number; biquadratic field

Zbl 0743.11061
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