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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