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PreprintCC BY 4.0arXiv

PAC fields and difference fields with generic automorphisms

  • Özlem Beyarslan
  • Piotr Kowalski
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Abstract

We study the existence of model companions after adjoining an automorphism to field theories in two complementary settings. For bounded PAC fields of characteristic zero containing all roots of unity and admitting a cyclic Galois quotient of prime order, a suitable choice of countably many constants gives a model-complete field theory which has no model companion after adding an automorphism fixing the named constants. This applies, in particular, when the absolute Galois group is free profinite or free pro-p of finite positive rank. Secondly, for every completion of ACFA in every characteristic, adjoining a second commuting automorphism has no model companion. The two explicit obstructions use a common ordered product of inertia automorphisms along an infinite orbit, combined respectively with an equivariant PAC completion and an algebraically closed difference-field embedding.

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Cite

BibTeXRIS
BibTeX
@misc{beyarslan2026fields,
  title = {{PAC fields and difference fields with generic automorphisms}},
  author = {Özlem Beyarslan and Piotr Kowalski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40251},
  eprint = {2609.40251},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40251},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - PAC fields and difference fields with generic automorphisms
AU  - Özlem Beyarslan
AU  - Piotr Kowalski
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40251
UR  - https://arxiv.org/abs/2609.40251
AB  - We study the existence of model companions after adjoining an automorphism to field theories in two complementary settings. For bounded PAC fields of characteristic zero containing all roots of unity and admitting a cyclic Galois quotient of prime order, a suitable choice of countably many constants gives a model-complete field theory which has no model companion after adding an automorphism fixing the named constants. This applies, in particular, when the absolute Galois group is free profinite or free pro-p of finite positive rank. Secondly, for every completion of ACFA in every characteristic, adjoining a second commuting automorphism has no model companion. The two explicit obstructions use a common ordered product of inertia automorphisms along an infinite orbit, combined respectively with an equivariant PAC completion and an algebraically closed difference-field embedding.
ER  -