利用小幂伽罗瓦群快速判定方程的可解性
Detecting Fast solvability of equations via small powerful Galois groups
摘要：
固定一个奇素数$p$，并设$F$是包含本原$p$第单位根的域。已知$p$ -刚性域$F$的特征在于最大$p$ -扩张$F(p)/F$的伽罗瓦群$G_F(p)$是可解群.本文给出了$p$ -刚性的一个新的刻画，即当绝对伽罗瓦群的两个基本正则群重合时，场$F$是$p$ -刚性的.这个条件进一步与解析$p$ -进群和某些伽罗瓦模有关。当$F$是$p$ -刚性的，我们还表明，它是可能的解决任何不可约多项式的根在$F[X]$的分裂领域在$F$有一个$p$ -权力度通过非嵌套的自由基。我们提供了新的直接证明遗传$p$ -刚性，连同$G_F(p)$的一些特征-包括一个完整的描述这样一个群体和行动的$F(p)$ -在$F$是$p$ -刚性的情况下。
英文摘要：
Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$ is a solvable group. We give a new characterization of $p$-rigidity which says that a field $F$ is $p$-rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic $p$-adic groups and to some Galois modules. When $F$ is $p$-rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in $F[X]$ whose splitting field over $F$ has a $p$-power degree via non-nested radicals. We provide new direct proofs for hereditary $p$-rigidity, together with some characterizations for $G_F(p)$ -- including a complete description for such a group and for the action of it on $F(p)$ -- in the case $F$ is $p$-rigid.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
