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Units and class numbers for a dihedral extension of $$\mathbb Q$$. (Unités et nombre de classes d’une extension diédrale de $$\mathbb Q$$.) (French) Zbl 0313.12002
Astérisque 24-25, 29-35 (1975).
Let $$K$$ be a dihedral extension of $$\mathbb Q$$ (i.e. $$\text{Gal}(K/\mathbb Q)\simeq D_P=\langle \sigma,\tau: \sigma^P=\tau^2=1, \sigma\tau \sigma =tau\rangle$$). Let $$P$$ be a prime. Let $$E_K$$ be the group of units of $$K$$ modulo roots of unity. Using results of M. P. Lee [Trans. Am. Math. Soc. 110, 213–231 (1964; Zbl 0126.05403)] on the structure of $$E_K$$, the author proves the class-number formulas:
(1) $$h_K=ah_L^2h_k/p^2$$, if $$K$$ is totally real;
(2) $$h_K=ah_L^2h_k/p$$, if $$K$$ is imaginary;
here, $$L$$ resp. $$K$$ is the subfield of $$K$$ fixed by $$\tau$$ resp. $$\sigma$$ and the unit index $$[E_K:E_L\cdot E_L^{\sigma}\cdot E_k]$$.
For the entire collection see [Zbl 0305.10003].
Reviewer: Ezra Brown

##### MSC:
 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions