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

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.

### 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
Full Text:

### 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
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