Number Theory Seminar: Amador Martin-Pizarro (Univ. of Freiburg)

On bounded automorphisms of fields with operators

Lascar showed that the group of automorphisms of the complex field which fix the algebraic closure of the prime field is simple. For this, he first showed that there are no non-trivial bounded automorphisms. An automorphism is bounded if there is a finite set A such that the image of every element b is algebraic over A together with b. The same result holds for a "universal" differentially closed field of characteristic zero, where we replace algebraic by differentially algebraic. Together with T. Blossier and C. Hardouin, we provided in https://arxiv.org/abs/1505.03669 a complete classification of bounded automorphisms in various fields equipped with operators, among others, for generic difference fields in all characteristics or for Hasse-Schmidt differential fields in positive characteristic.

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