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تسجيل مستخدم جديد$3$-Principalization over $S_3$-fields
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Let $pequiv 1,(mathrm{mod},9)$ be a prime number and $zeta_3$ be a primitive cube root of unity. Then $mathrm{k}=mathbb{Q}(sqrt[3]{p},zeta_3)$ is a pure metacyclic field with group $mathrm{Gal}(mathrm{k}/mathbb{Q})simeq S_3$. In the case that $mathrm{k}$ possesses a $3$-class group $C_{mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $mathrm{k}_3^{(infty)}$ of $mathrm{k}$ are drawn.
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