We determine the 3-class groups of \({\mathbb {Q}}(\root 3 \of {p})\) and \(K={\mathbb {Q}}(\root 3 \of {p},\sqrt{-3})\) when \(p\equiv 4,7\bmod 9\) is a prime and 3 is a cube modulo p. This confirms a conjecture made by Barrucand-Cohn, and proves the last remaining case of a conjecture of Lemmermeyer on the 3-class group of K.

中文翻译：

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$$\mathbb {Q}(\root 3 \of {p})$$ Q ( p 3 ) 的三类群及其正态闭包

我们确定了\({\mathbb {Q}}(\root 3 \of {p})\)和\(K={\mathbb {Q}}(\root 3 \of {p} ,\sqrt{-3})\)当\(p\equiv 4,7\bmod 9\)是素数并且 3 是模p的立方体。这证实了 Barrucand-Cohn 的猜想，并证明了 Lemmermeyer 对K的 3 类群的猜想的最后一个案例。