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We investigate definability in henselian fields. Specifically, we are interested in those sets and substructures that are existentially definable or definable with `few' parameters. Our general approach is to use various versions of henselianity to understand the `local structure' of these definable sets. The fields in which we are most interested are those of positive characteristic, for example the local fields Fq((t)), but many of our methods and results also apply to p-adic and real closed fields. In positive characteristic we have to deal with inseparable field extensions and we develop the method of Λ-closure to `translate' inseparable field extensions into separable ones. In the first part of the thesis we focus on existentially definable sets, which are projections of algebraic sets. Our main tool is the Implicit Function Theorem (for polynomials) which is equivalent to t-henselianity, by work of Prestel and Ziegler. This enables us to prove that existentially definable sets are `large' in various senses. Using the Implicit Function Theorem, we also obtain a nonuniform local elimination of the existential quantifier. The non-uniformity and local character of this result at present forms an obstacle to full quantifier-elimination. From these technical statements we can deduce characterisations of, for example, existentially definable subfields and existentially definable transcendentals. We prove that a dense, regular extension of t-henselian fields is existentially closed which, in particular, implies the old result of Ershov that Fp(t)h ≤Ǝ Fp((t)). Using the existential closedness of large fields in henselian fields, we are able to apply many of these results to large fields. This answers questions for imperfect large fields that were answered in the perfect case by Fehm. In the second part of the thesis, we work with power series fields F((t)) and subsets which are F- definable (and not contained in F). We use a `hensel-like' lemma to characterise F-orbits of (singleton) elements of F((t)). It turns out that all such orbits are Ǝ-t-definable. Consequently, we may apply our earlier results about existentially definable subsets to F-definable subsets. We can use this to characterise F-definable subfields of F((t)). As a further corollary, we obtain an Ǝ-0̸-definition of Fp[[t]] in Fp((t)). |
