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Register a new user$3$-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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We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
If the $ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $sX$ is regular, one has a congruence $|sX(k)|equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.
Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
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