Galois Groups Over Nonrigid Fields

Details Galois Groups Over Nonrigid Fields Galois Groups Over Nonrigid Fields

2000-10-20

arxiv.org/abs/math/0010201v1

Mathematics

Number Theory

Scientific paper

Let $F$ be a field with characteristic $\neq 2$. We show that $F$ is a nonrigid field if and only if certain small 2-groups occur as Galois groups over $F$. These results provide new "automatic realizability" results for Galois groups over $F$. The groups we consider demonstrate the inequality of two particular metabelian 2-extensions of $F$ which are unequal precisely when $F$ is a nonrigid field. Using known results on connections between rigidity and existence of certain valuations, we obtain Galois-theoretic criteria for the existence of these valuations.

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