Small Galois groups that encode valuations

Mathematics – Number Theory

Scientific paper

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Final version. To appear in Acta Arithmetica

Scientific paper

Let $p$ be a prime number and let $F$ be a field containing a root of unity
of order $p$. We prove that a certain very small canonical Galois group
$(G_F)_{[3]}$ over $F$ encodes the valuations on $F$ whose value group is not
$p$-divisible and which satisfy a variant of Hensel's lemma.

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