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Submission: 2002, Jan 20
Let $F$ be a field with $\cha F \neq 2$. We show that there are two groups of order 32, respectively 64, such that a field $F$ with $\cha F \neq 2$ is nonrigid if and only if at least one of the two groups is realizable as a Galois group over $F$. The realizability of those groups turns out to be equivalent to the realizability of certain quotients (of order 16, respectively 32). Using known results on connections between rigidity and existence of certain valuations, we obtain new Galois-theoretic criteria for the existence of these valuations.
2000 Mathematics Subject Classification:
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