Abelian subgroups of pro-$p$ Galois groups
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- by Antonio José Engler and Jochen Koenigsmann PDF
- Trans. Amer. Math. Soc. 350 (1998), 2473-2485 Request permission
Abstract:
It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois $p$-extension of a field $F$ (where $p$ is an odd prime) arise from $p$-henselian valuations with non-$p$-divisible value group, provided $\# (\dot {F}/\dot {F}^{p})\geq p^{2}$ and $F$ contains a primitive $p$-th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.References
- Jón Kr. Arason, Richard Elman, and Bill Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra 110 (1987), no. 2, 449–467. MR 910395, DOI 10.1016/0021-8693(87)90057-3
- Eberhard Becker, Hereditarily-Pythagorean fields and orderings of higher level, Monografías de Matemática [Mathematical Monographs], vol. 29, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978. MR 527118
- S. V. Bredihin, Ju. L. Eršov, and V. E. Kal′neĭ, Fields with two total orderings, Mat. Zametki 7 (1970), 525–536 (Russian). MR 266910
- Ludwig Bröcker, Characterization of fans and hereditarily Pythagorean fields, Math. Z. 151 (1976), no. 2, 149–163. MR 422233, DOI 10.1007/BF01213992
- Otto Endler, On henselizations of valued fields, Bol. Soc. Brasil. Mat. 4 (1973), no. 2, 97–109. MR 417143, DOI 10.1007/BF02584715
- Otto Endler and Antonio José Engler, Fields with Henselian valuation rings, Math. Z. 152 (1977), no. 2, 191–193. MR 427284, DOI 10.1007/BF01214188
- Ido Efrat, Abelian subgroups of pro-$2$ Galois groups, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1031–1035. MR 1242081, DOI 10.1090/S0002-9939-1995-1242081-X
- Antonio J. Engler, Fields with two incomparable Henselian valuation rings, Manuscripta Math. 23 (1977/78), no. 4, 373–385. MR 463148, DOI 10.1007/BF01167696
- Claude Carlet, The divisors of $x^{2^m}+x$ of constant derivatives and degree $2^{m-2}$, SIAM J. Discrete Math. 7 (1994), no. 2, 238–244. MR 1271996, DOI 10.1137/S0895480192229133
- Yoon Sung Hwang and Bill Jacob, Brauer group analogues of results relating the Witt ring to valuations and Galois theory, Canad. J. Math. 47 (1995), no. 3, 527–543. MR 1346152, DOI 10.4153/CJM-1995-029-4
- Jochen Koenigsmann, $p$-Henselian fields, Manuscripta Math. 87 (1995), no. 1, 89–99. MR 1329442, DOI 10.1007/BF02570463
- Jochen Koenigsmann, From $p$-rigid elements to valuations (with a Galois-characterization of $p$-adic fields), J. Reine Angew. Math. 465 (1995), 165–182. With an appendix by Florian Pop. MR 1344135, DOI 10.1515/crll.1995.465.165
- Florian Pop, On Grothendieck’s conjecture of birational anabelian geometry, Ann. of Math. (2) 139 (1994), no. 1, 145–182. MR 1259367, DOI 10.2307/2946630
- Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847, DOI 10.1007/BFb0101548
- Roger Ware, When are Witt rings group rings? II, Pacific J. Math. 76 (1978), no. 2, 541–564. MR 568320
- Roger Ware, Galois groups of maximal $p$-extensions, Trans. Amer. Math. Soc. 333 (1992), no. 2, 721–728. MR 1061780, DOI 10.1090/S0002-9947-1992-1061780-8
Additional Information
- Antonio José Engler
- Affiliation: IMECC-UNICAMP, Caixa Postal 6065, 13083-970, Campinas, SP, Brasil
- Email: engler@ime.unicamp.br
- Jochen Koenigsmann
- Affiliation: Fakulta̋t fűr Mathematik, Universita̋t Konstanz, Postfach 5560, D-78434 Konstanz, Germany
- Email: jochen.koenigsmann@uni-konstanz.de
- Received by editor(s): December 20, 1995
- Received by editor(s) in revised form: September 11, 1996
- Additional Notes: The contents of this paper were developed while the first author enjoyed the hospitality of Konstanz University supported by GMD-CNPq.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 2473-2485
- MSC (1991): Primary 12F10; Secondary 12J20
- DOI: https://doi.org/10.1090/S0002-9947-98-02063-7
- MathSciNet review: 1451599