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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.

MSC:

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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