Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A generalization of Marshall's equivalence relation

Author: Ido Efrat
Journal: Trans. Amer. Math. Soc. 358 (2006), 2561-2577
MSC (2000): Primary 12E30; Secondary 12J15, 19C99, 12J99
Published electronically: September 22, 2005
MathSciNet review: 2204044
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For $p$ prime and for a field $F$ containing a root of unity of order $p$, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of $F^{\times }$ of index $p$. The equivalence classes then correspond to free pro-$p$ factors of the maximal pro-$p$ Galois group of $F$. We generalize to this setting results of Jacob on the maximal pro-$2$ Galois group of a Pythagorean field.

References [Enhancements On Off] (What's this?)

  • [B] E. Becker, Euklidische Körper und euklidische Hüllen von Körpern, J. reine angew. Math. 268-269 (1974), 41-52. MR 0354625 (50:7103)
  • [Bo] N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972. MR 0360549 (50:12997)
  • [Br] L. Bröcker, Characterization of fans and hereditarily pythagorean fields, Math. Z. 151 (1976), 149-163. MR 0422233 (54:10224)
  • [C1] T.C. Craven, Characterizing reduced Witt rings of fields, J. Algebra 53 (1978), 68-77. MR 0480332 (58:505)
  • [C2] T.C. Craven, Characterizing reduced Witt rings II, Pacific J. Math. 80 (1979), 341-349. MR 0539420 (80i:10025)
  • [E1] I. Efrat, Free product decompositions of Galois groups over pythagorean fields, Comm. Algebra 21 (1993), 4495-4511. MR 1242845 (95a:12003)
  • [E2] I. Efrat, Orderings, valuations and free products of Galois groups, In: Séminaire de Structures Algébriques Ordonnées, Lecture Notes No. 54, University of Paris VII, 1995.
  • [E3] I. Efrat, Free pro-$p$ product decompositions of Galois groups, Math. Z. 225 (1997), 245-261. MR 1464929 (98i:12004)
  • [E4] I. Efrat, Pro-$p$ Galois groups of algebraic extensions of $\mathbb{Q}$, J. Number Theory 64 (1997), 84-99. MR 1450486 (98i:11096)
  • [E5] I. Efrat, Finitely generated pro-$p$ Galois groups of $p$-henselian fields, J. Pure Appl. Algebra 138 (1999), 215-228. MR 1691472 (2000e:12011)
  • [E6] I. Efrat, Finitely generated pro-$p$ absolute Galois groups over global fields, J. Number Theory 77 (1999), 83-96. MR 1695702 (2000c:12007)
  • [E7] I. Efrat, Pro-$p$ Galois groups of function fields over local fields, Comm. Algebra 28 (2000), 2999-3021. MR 1757442 (2001g:12001)
  • [E8] I. Efrat, A Hasse principle for function fields over PAC fields, Israel J. Math. 122 (2001), 43-60. MR 1826490 (2002a:14018)
  • [E9] I. Efrat, Demuskin fields with valuations, Math. Z. 243 (2003), 333-353.MR 1961869 (2004d:11116)
  • [EH] I. Efrat and D. Haran, On Galois groups over pythagorean and semi-real closed fields, Israel J. Math. 85 (1994), 57-78. MR 1264339 (94m:12002)
  • [FeV] I.B. Fesenko and S.V. Vostokov, Local Fields and their Extensions - A Constructive Approach, AMS, Providence, Rhode Island, 1993. MR 1218392 (94d:11095)
  • [FJ] M. Fried and M. Jarden, Field Arithmetic, Springer, Heidelberg, 1986.MR 0868860 (89b:12010)
  • [H] D. Haran, On closed subgroups of free products of profinite groups, Proc. London Math. Soc. 55 (1987), 266-298. MR 0896222 (88i:20047)
  • [HR] W.N. Herfort and L. Ribes, Torsion elements and centralizers in free products of profinite groups, J. reine angew. Math. 358 (1985), 155-161.MR 0797680 (86k:20024)
  • [HwJ] Y.S. Hwang and B. Jacob, Brauer group analogues of results relating the Witt ring to valuations and Galois theory, Canad. J. Math. 47 (1995), 527-543. MR 1346152 (97a:12004)
  • [J] B. Jacob, On the structure of Pythagorean fields, J. Algebra 68 (1981), 247-267. MR 0608534 (82g:12020)
  • [JWr1] B. Jacob and R. Ware, A recursive description of the maximal pro-$2$ Galois group via Witt rings, Math. Z. 200 (1989), 379-396. MR 0978598 (90b:11127)
  • [JWr2] B. Jacob and R. Ware, Realizing dyadic factors of elementary type Witt rings and pro-$2$ Galois groups, Math. Z. 208 (1991), 193-208. MR 1128705 (92h:11032)
  • [Jr] M. Jarden, Intersections of local algebraic extensions of a Hilbertian field, NATO Adv. Sci. Inst. Ser. C, ``Generators and Relations in Groups and Geometries'', Ed.: Barlotti et. al., pp. 343-405, 1991.MR 1206921 (94c:12003)
  • [JP] C.U. Jensen and A. Prestel, Finitely generated pro-p-groups as Galois groups of maximal $p$-extensions of function fields over $\mathbb{Q}_{q}$, manusc. math. 90 (1997), 225-238. MR 1391210 (97f:11091)
  • [L] T.Y. Lam, Orderings, valuations and quadratic forms, Conf. Board of the Mathematical Sciences 52, AMS 1983. MR 0714331 (85e:11024)
  • [Ma1] M. Marshall, Spaces of orderings IV, Canad. J. Math. 32 (1980), 603-627. MR 0586979 (81m:10035)
  • [Ma2] M. Marshall, Abstract Witt Rings, Queen's Pap. Pure Appl. Math. 57, Kingston, 1980. MR 0674651 (84b:10032)
  • [Ma3] M. Marshall, Spaces of Orderings and Abstract Real Spectra, Lect. Notes Math. 1636, Springer, Berlin-Heidelberg, 1996. MR 1438785 (98b:14041)
  • [Ma4] M. Marshall, The elementary type conjecture in quadratic form theory, Cont. Math. 344 (2004), 275-293. MR 2060204 (05b:11046)
  • [Mel] O.V. Melnikov, Subgroups and homologies of free products of profinite groups, Izvestiya Akad. Nauk SSSR, Ser. Mat. 53 (1989), 97-120 (Russian); Math. USSR Izvestiya 34 (1990), 97-119 (English translation). MR 0992980 (91b:20033)
  • [Mer] J. Merzel, Quadratic forms over fields with finitely many orderings, Contemporary Math. 8 (1982), 185-229. MR 0653183 (83j:10021)
  • [Mi] J. Minác, Galois groups of some 2-extensions of ordered fields, C.R. Math. Rep. Acad. Sci. Canada 8 (1986), 103-108.MR 0831786 (88k:12003a)
  • [P] F. Pop, Galoissche Kennzeichnung $p$-adisch abgeschlossener Körper, J. reine angew. Math. 392 (1988), 145-175. MR 0965062 (89k:12014)
  • [S1] J.-P. Serre, Local Fields, Springer, Berlin, 1979. MR 0554237 (82e:12016)
  • [S2] J.-P. Serre, Galois Cohomology, Springer Monographs in Mathematics, Springer, Berlin Heidelberg, 1997. MR 1466966 (98g:12007)
  • [Wd] A.R. Wadsworth, $p$-Henselian fields: $K$-theory, Galois cohomology, and graded Witt rings, Pac. J. Math. 105 (1983), 473-496. MR 0691616 (84m:12026)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 12E30, 12J15, 19C99, 12J99

Retrieve articles in all journals with MSC (2000): 12E30, 12J15, 19C99, 12J99

Additional Information

Ido Efrat
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel

Received by editor(s): September 27, 2003
Received by editor(s) in revised form: June 20, 2004
Published electronically: September 22, 2005
Additional Notes: This research was supported by the Israel Science Foundation grant No. 8008/02–1
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society