Existentially generated subfields of large fields

Anscombe, Sylvy orcid iconORCID: 0000-0002-9930-2804 (2019) Existentially generated subfields of large fields. Journal of Algebra, 517 . pp. 78-94. ISSN 0021-8693

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Official URL: https://doi.org/10.1016/j.jalgebra.2018.09.021


We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated.Let L be a large field of characteristic exponent p, and let E \subseteq L be an infinite existentially
generated subfield. We show that E contains L^(p^n), the p^n-th powers in L, for some n < ω.

This generalises a result of Fehm, which shows E = L, under the assumption that L is perfect. Our method is to first study existentially generated subfields of henselian fields. Since L is existentially closed in the henselian field L((t)), our result follows.

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