Abelian subgroups of pro-2 Galois groups

Efrat, Ido

doi:10.1090/S0002-9939-1995-1242081-X

Abelian subgroups of pro-$2$ Galois groups
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Proc. Amer. Math. Soc. 123 (1995), 1031-1035 Request permission

Abstract:

Let $a(K)$ be the maximal cardinality $|I|$ such that $\mathbb {Z}_2^I$ is a closed subgroup of the maximal pro-2 Galois group of a field K. We prove estimates on $a(K)$ conjectured by Ware.

References

Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549
Ludwig Bröcker, Characterization of fans and hereditarily Pythagorean fields, Math. Z. 151 (1976), no. 2, 149–163. MR 422233, DOI 10.1007/BF01213992
Ido Efrat, Free product decompositions of Galois groups over Pythagorean fields, Comm. Algebra 21 (1993), no. 12, 4495–4511. MR 1242845, DOI 10.1080/00927879308824813
Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379
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
Bill Jacob and Roger Ware, A recursive description of the maximal pro-$2$ Galois group via Witt rings, Math. Z. 200 (1989), no. 3, 379–396. MR 978598, DOI 10.1007/BF01215654
Tsit Yuen Lam, Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 714331, DOI 10.1090/cbms/052
Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063, DOI 10.1007/978-3-642-69971-9
Jean-Pierre Serre, Cohomologie galoisienne, Lecture Notes in Mathematics, No. 5, Springer-Verlag, Berlin-New York, 1965 (French). With a contribution by Jean-Louis Verdier; Troisième édition, 1965. MR 0201444
Roger Ware, When are Witt rings group rings? II, Pacific J. Math. 76 (1978), no. 2, 541–564. MR 568320
Roger Ware, Stability in Witt rings and abelian subgroups of pro-$2$-Galois groups, Rocky Mountain J. Math. 19 (1989), no. 3, 985–995. Quadratic forms and real algebraic geometry (Corvallis, OR, 1986). MR 1043266, DOI 10.1216/RMJ-1989-19-3-985