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Are there any restrictions on the ground field over which the implicit function theorem holds? In particular, does the theorem hold over function fields like $F_q((1/t))$?

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The field $\mathbb{F}_q((1/t))$ is not a "function field", at least not according to my philosophy. Perhaps you could specify what you mean by a function field and also what statement counts as "the implicit function theorem" over your function fields? – Pete L. Clark Mar 30 '11 at 17:22

There is an implicit function theorem valid for any non-Archimedean field. See Theorem 2.2.1 in J.-I. Igusa, An introduction to the theory of local zeta functions. AMS, 2000.

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Essentially, it is explained in the answer to this math.stackexchange question that the implicit function theorem is equivalent to the validity of Hensel's lemma. The fields where the implicit function theorem holds are called henselian fields.

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