Henselian implies large. (English) Zbl 1220.12001

A field \(K\) is called large if every smooth \(K\)-curve which has a \(K\)-rational point has an infinity of \(K\)-rational points. The author proves that the quotient field of a domain which is Henselian with respect to a nontrivial ideal is a large field. Every nontrivial finite split embedding problem for the absolute Galois group of a Hilbertian large Krull field \(K\) has \(|K|\) independent and totally ramified proper solutions. Let \((R,{\mathfrak m})\) be an excellent two dimensional Henselian local ring with separably closed residue field \(k\) such that the quotient field \(K\) of \(R\) has the same characteristic as \(k\). If \(|k|<|R|\), suppose that there exists \(x\in {\mathfrak m}\) such that \(k[[x]]\subset R\). Then the absolute Galois group of the maximal abelian extension \(K^{\text{ab}}\) of \(K\) is profinite free on \(|K^{\text{ab}}|\) generators.


12F12 Inverse Galois theory
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
12E30 Field arithmetic
12F10 Separable extensions, Galois theory
12G10 Cohomological dimension of fields
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[1] L. Moret-Bailly, ”\(R\)-équivalence simultanée de torseurs: un complément à l’article de P. Gille,” J. Number Theory, vol. 91, iss. 2, pp. 293-296, 2001. · Zbl 1076.14531
[2] J. Colliot-Thélène, ”Rational connectedness and Galois covers of the projective line,” Ann. of Math., vol. 151, iss. 1, pp. 359-373, 2000. · Zbl 0990.12003
[3] J. Colliot-Thélène, M. Ojanguren, and R. Parimala, ”Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes,” in Algebra, arithmetic and geometry, Parts I, II, Bombay: Tata Inst. Fund. Res., 2002, vol. 16, pp. 185-217. · Zbl 1055.14019
[4] P. Dèbes and B. Deschamps, ”The regular inverse Galois problem over large fields,” in Geometric Galois Actions II, Schneps, L. and Lochak, P., Eds., Cambridge: Cambridge Univ. Press, 1997, vol. 243, pp. 119-138. · Zbl 0905.12004
[5] W. Geyer, ”Galois groups of intersections of local fields,” Israel J. Math., vol. 30, iss. 4, pp. 382-396, 1978. · Zbl 0383.12014
[6] D. Harbater, ”Patching and Galois theory,” in Galois Groups and Fundamental Groups, Cambridge: Cambridge Univ. Press, 2003, vol. 41, pp. 313-424. · Zbl 1071.14029
[7] D. Harbater, ”On function fields with free absolute Galois groups,” J. Reine Angew. Math., vol. 632, pp. 85-103, 2009. · Zbl 1192.12004
[8] D. Harbater and K. F. Stevenson, ”Local Galois theory in dimension two,” Adv. Math., vol. 198, iss. 2, pp. 623-653, 2005. · Zbl 1104.12003
[9] J. Kollár, ”Rationally connected varieties over local fields,” Ann. of Math., vol. 150, iss. 1, pp. 357-367, 1999. · Zbl 0976.14016
[10] J. Lafon, ”Anneaux henséliens,” Bull. Soc. Math. France, vol. 91, pp. 77-107, 1963. · Zbl 0126.06601
[11] E. Paran, ”Split embedding problems of complete domains,” Ann. of Math., vol. 170, pp. 899-914, 2009. · Zbl 1247.12005
[12] B. Poonen and E. Pop, ”First order characterization of function field invariants over large fields,” in Model Theory with Applications to Algebra and Analysis, Chatzidakis, Macphersons, Pillays, and Wilkie, Eds., Cambridge Univ. Press, 2008, vol. 350, pp. 255-272. · Zbl 1174.03015
[13] F. Pop, ”Embedding problems over large fields,” Ann. of Math., vol. 144, iss. 1, pp. 1-34, 1996. · Zbl 0862.12003
[14] L. Positselski, ”Koszul property and Bogomolov’s conjecture,” Int. Math. Res. Not., vol. 31, pp. 1901-1936, 2005. · Zbl 1160.19301
[15] R. Weissauer, ”Der Hilbertsche Irreduzibilitätssatz,” J. Reine Angew. Math., vol. 334, pp. 203-220, 1982. · Zbl 0477.12029
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