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PreprintarXiv licencearXiv

Hilbertian Fields and Profinite Iterated Monodromy Groups Over Transcendental Elements with Specializations of Infinite Index

  • Daniel Gotshall
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Abstract

Given a field L and polynomial f(x)/L, one can consider the indices of the profinite iterated monodromy groups formed over the ground field within the profinite iterated monodromy groups over elements transcendental over L. For any countable Hilbertian field k, we show random Galois fixed field extensions almost surely make the indices infinite over all ground field elements, with certain conditions on f(x). This is a generalization of a result in arXiv:2012.03076 by Dittmann and Kadets.

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Cite

BibTeXRIS
BibTeX
@misc{gotshall2026hilbertian,
  title = {{Hilbertian Fields and Profinite Iterated Monodromy Groups Over Transcendental Elements with Specializations of Infinite Index}},
  author = {Daniel Gotshall},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40017},
  eprint = {2610.40017},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40017},
}
RIS
TY  - UNPB
TI  - Hilbertian Fields and Profinite Iterated Monodromy Groups Over Transcendental Elements with Specializations of Infinite Index
AU  - Daniel Gotshall
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40017
UR  - https://arxiv.org/abs/2610.40017
AB  - Given a field L and polynomial f(x)/L, one can consider the indices of the profinite iterated monodromy groups formed over the ground field within the profinite iterated monodromy groups over elements transcendental over L. For any countable Hilbertian field k, we show random Galois fixed field extensions almost surely make the indices infinite over all ground field elements, with certain conditions on f(x). This is a generalization of a result in arXiv:2012.03076 by Dittmann and Kadets.
ER  -