## On the parity of the class number of the field $$\mathbb Q(\sqrt p,\sqrt q,\sqrt r)$$.(English)Zbl 0913.11046

The author determines the parity of the class number of the octic field $$K= \mathbb Q(\sqrt{p}, \sqrt{q}, \sqrt{r})$$, where $$p,q,r$$ are different primes and $$p\equiv q\equiv r\equiv 1\bmod 4$$. The author establishes that if the Legendre symbols $$(p/q)= (p/r)= (q/r)=1$$ or $$(p/q)= (q/r)= 1$$, $$(p/r)=-1$$ then $$K$$ has even 2-class number (i.e. the class number of the 2-Sylow subgroup of the ideal class group of $$K$$). If $$(p/q)=1$$, $$(p/r)= (q/r)=-1$$ then the parity of the class number of $$K$$ is the same as the parity of the class number of the biquadratic field $$\mathbb Q (\sqrt{p}, \sqrt{q})$$; if $$(p/q)= (p/r)= (q/r)=-1$$ then the parity of the class number of $$K$$ is determined by a combination of specific congruence relations and related Legendre symbols. In addition, the author shows that if $$(p/q)=(p/r)=(q/r)=1$$, then the 2-class number of $$K$$ is greater than or equal to twice the product of the 2-class numbers of the three quadratic subfields of $$K$$.
In order to prove these results the author utilizes the results and techniques of R. Kučera [J. Number Theory 52, 43–52 (1995; Zbl 0852.11065)] for the biquadratic field $$\mathbb Q (\sqrt{p}, \sqrt{q})$$ where $$p$$ and $$q$$ are different primes and $$p\equiv q\equiv 1\bmod 4$$. The techniques utilized in the present paper revolve around the cyclotomic units of $$K$$, the fundamental units in the quadratic subfields of $$K$$, the fixed Dirichlet characters modulo $$t$$ of order 4, where $$t\neq 1$$ divides $$pqr$$, and crossed homomorphisms from $$G= \text{Gal} (K/\mathbb Q)$$ to $$K$$. Specifically, the formulation for the parity of the class number depends upon units of the form $$B_{pq}= \prod_{a\in M_{pq}} ({\mathcal E}_{pq}^a- {\mathcal E}_{pq}^{-a})$$, where $$M_{pq}= \{a\in \mathbb Z\mid 0<a< pq$$, $$(a/q)=1\}$$, $$\psi_p(a)\in R_p$$, where the symbols are defined as follows: Let $$E$$ be the group of units in $$K= \mathbb Q (\sqrt{p}, \sqrt{q}, \sqrt{r})$$, $${\mathcal E}_n= e^{\frac{2\pi i}{n} \frac{1+n}{2}}$$ for any positive integer $$n$$; for any prime $$t\equiv 1\bmod 4$$ let $$b_t,c_t$$ be integers such that $$t-1= 2^{b_t}c_t$$, where $$2\nmid c_t$$ and $$b_t\geq 2$$, and let $$\psi_t$$ denote a fixed Dirichlet character modulo $$t$$ of order $$2^{b_t}$$; let $$R_t= \{p_t^j\mid 0\leq j<2^{b_t-2}\}$$ where $$p_t= e^{4\pi ict/t-1}$$ is a primitive $$2^{b_t-1}$$th root of unity. A basic result used to establish the parity of the class number $$h$$ of $$K$$ states that $$h$$ is even if and only if there are $$\chi_p, \chi_q, \chi_r, \chi_{pq},\chi_{qr}, \chi_{pqr}\in \{0,1\}$$ such that $n=| {\mathcal E}_p^{\chi_p} {\mathcal E}_q^{\chi_q}{\mathcal E}_r^{\chi_r} {\mathcal B}_{pq}^{\chi_{pq}} {\mathcal B}_{pr}^{\chi_{pr}} {\mathcal B}_{qr}^{\chi_{qr}} {\mathcal B}_{pqr}^{\chi_{pqr}} |\neq 1$ is a square in $$E$$ $$({\mathcal E}_s$$ denotes the fundamental unit in the quadratic field $$\mathbb Q (\sqrt{s})$$ and $$\chi_s$$ denotes a fixed Dirichlet character modulo $$S$$ of order 4). For example, in order to establish the parity criteria when $$(p/q)= (q/r)=1$$, $$(p/r)=-1$$, the author demonstrates that a necessary condition for $$n$$ being a square in $$E$$ is that $$\chi_q(r)\cdot \chi_r(q)= \chi_q(p)\cdot \chi_p(q)$$.

### MSC:

 11R29 Class numbers, class groups, discriminants 11R20 Other abelian and metabelian extensions 11R27 Units and factorization

Zbl 0852.11065
Full Text:

### References:

  Kučera, R., On the parity of the class number of a biquadratic field, J. number theory, 52, 43-52, (1995) · Zbl 0852.11065  Kučera, R., On the Stickelberger ideal and circular units of a compositum of quadratic fields, J. number theory, 56, 139-166, (1996) · Zbl 0840.11044
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