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Stark units and Hilbert class fields. (Unités de Stark et corps de classes de Hilbert.) (French) Zbl 0871.11080

A conjecture of H. M. Stark [Bull. Am. Math. Soc. 83, 1072-1074 (1977; Zbl 0378.12007)] states that for certain abelian extensions \(K/k\) the derivative of an Artin \(L\)-function determines a unit \(\varepsilon\) of \(K\). The author uses this conjecture to prove if \(k\) is a totally real number field of degree \(n\geq 3\) and \(\eta\) is an integer of \(k\) which has exactly one real conjugate counting multiplicities, and \(K=H(\sqrt{\eta})\) where \(H\) is the Hilbert class field of \(k\), then \(K=k(\varepsilon)=\mathbb{Q}(\varepsilon)\), where \(\varepsilon\) is a unit satisfying Stark’s conjecture.
Moreover, a generator \(\alpha\) for \(H/\mathbb{Q}\) is easily determined from \(\varepsilon\). Two examples are given where the Hilbert class field is determined for totally real cubic fields. One of these has class number 3 while the other one has class number 5.

MSC:

11R37 Class field theory
11R27 Units and factorization

Citations:

Zbl 0378.12007
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