What is the free field? The talk is intended to answer this question, and to show some links with automata theory and linear recursive sequences. Roughly speaking, the free field is a noncommutative analogue of the field of rational functions. It is not easy to construct, since the construction with fractions does not work. One possibility is to embed it into the ring of Malcev-Neumann series on the free group (Lewin). Another one is to develop, as Paul Cohn does, a noncommutative theory of localization. Another one is to view it as the set of rational expressions modulo the universal identities (Amitsur).
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