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The 3-class groups of \(\mathbb{Q}(\sqrt[3]{p})\) and its normal closure. (English) Zbl 1496.11134

Let \(S\) be a finite set of prime ideals of a number field. In the paper under review the authors use the \(S\)-version of Chevalley’s ambiguous class number formula to prove that if \(p\) is a prime such that \(p\equiv 4,7\pmod 9\), and the cubic residue symbol \(\left (\frac 3 p\right )_3\) has value 1, then the 3-class group of \(F=\mathbb Q(\sqrt[3]{p})\) is isomorphic to \(\mathbb Z/3\mathbb Z\), and the 3-class group of its normal closure \(K=\mathbb Q(\sqrt[3]{p}, \sqrt{-3})\) is isomorphic to \((\mathbb Z/3\mathbb Z)^2\). This confirms a conjecture made in [P. Barrucand and H. Cohn, J. Number Theory 2, 7–21 (1970; Zbl 0192.40001)] and completes the proof of a conjecture of Lemmermeyer on the 3-class group of \(K\). The authors also present some consequences of their result regarding the group of units of \(K\) and the norm equation \( \mathbf N_{F/\mathbb Q}(x)=3\).

MSC:

11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions

Citations:

Zbl 0192.40001
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References:

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