A valuation on a field is a homomorphic mapping from the multiplicative group of invertible elements of a field into an ordered abelian group. If it maps into the additive group of integers, it is called discrete. By Chevalley’s theorem, every valuation on a field extends to any field extension. Henselian valued fields are those for which valuation extends uniquely to any algebraic field extension. For example, complete discrete valued fields are Henselian.
Two structures of a given language are elementary equivalent if and only if every sentence in this language holds in the first structure if and only if it also holds in the second. All isomorphic structures are elementary equivalent, but the converse is not true in general. The Ax-Kochen-Eršov theorem explains when any two Henselian valued fields are elementary equivalent. As a consequence, a sentence holds in the field of $p$-adic numbers ${\mathbb Q}_p$ for almost all primes $p$ if and only if it holds in the field of Laurent series ${\mathbb Z}_p((t))$ for almost all primes $p$.
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