On finite intersections of ''Henselian valued'' fields (Q1069991)

The author considers fields \(K=H_ 1\cap \cdots \cap H_ n\) where \(H_ i\) is a Henselian extension of \(K\) with respect to a valuation ring \(A_ i\) for \(1\leq i\leq m\) and \(H_ i\) is a real closure of \(K\) with respect to a positive cone \(A_ i\) for \(m+1\leq i\leq n\). It is assumed that different \(A_ i's\) induce different topologies on K. In Section 1 among others a generalization of a theorem of Neukirch is given: \(K\) is dense in each \(H_ i\) (1\(\leq i\leq n).\) In Section 2 the author studies ''maximal'' intersections \(K=H_ 1\cap \cdots \cap H_ n\), i.e. for all \((\sigma_ 1,\ldots,\sigma_ n)\in Aut(\tilde K/K)^ n\), K equals also the intersection \(\sigma_ 1(H_ 1)\cap \cdots \cap \sigma_ n(H_ n)\). Maximal intersections are characterized by the following generalization of Hensel's lemma: Every monic and irreducible polynomial \(p\in (A_ 1\cap \cdots \cap A_ m)[X]\) which has a simple zero in every residue field \(\bar A_ i\) (1\(\leq i\leq m)\) and changes sign with respect to each \(A_ i\) \((m+1\leq i\leq n)\) is linear. It is also proved that many maximal intersections exist. In Section 3 it is shown that the absolute Galois group of a maximal intersection \(K\) which also satisfies a certain local-global principle for absolute irreducible K-varieties, is the free pro-p-product of the absolute Galois group \(G_ i\) of \(H_ i\) (1\(\leq i\leq n)\), provided all \(G_ i\) are pro-p-groups.