PreprintarXiv licencearXiv
Ab-faithfulness of absolute Galois groups in number theory
Abstract
We prove that the absolute Galois group of every discretely valued field of mixed characteristic is ab-faithful. We also prove ab-faithfulness for higher local fields of positive dimension and give further examples using finite split embedding problems. Finally, we give a formal Hilbert~90 criterion for center-freeness and slimness of finite-step solvable quotients. We apply this criterion to ab-faithful absolute Galois groups and obtain slimness for several classes of fields, including higher local fields of positive dimension.
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BibTeX
@misc{yamaguchi2026abfaithfulness,
title = {{Ab-faithfulness of absolute Galois groups in number theory}},
author = {Naganori Yamaguchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05634},
eprint = {2610.05634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05634},
}
RIS
TY - UNPB
TI - Ab-faithfulness of absolute Galois groups in number theory
AU - Naganori Yamaguchi
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05634
UR - https://arxiv.org/abs/2610.05634
AB - We prove that the absolute Galois group of every discretely valued field of mixed characteristic is ab-faithful. We also prove ab-faithfulness for higher local fields of positive dimension and give further examples using finite split embedding problems. Finally, we give a formal Hilbert~90 criterion for center-freeness and slimness of finite-step solvable quotients. We apply this criterion to ab-faithful absolute Galois groups and obtain slimness for several classes of fields, including higher local fields of positive dimension.
ER -