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Some analogue of Hilbert’s irreducibility theorem and the distribution of algebraic number fields. (English) Zbl 0747.14001

Let \(t_ 1,\ldots,t_ n\) be algebraically independent over \(\mathbb{Q}\), \(k=\mathbb{Q}(t_ 1,\ldots,t_ n)\) and \(f(x,t)=x^ n+t_ 1x^{n- 1}+\cdots+t_ n\). Let \(K\) be the splitting field of \(f(x,t)\) over \(k\). \(G(K/k)\) is isomorphic to \(S_ n\). Let \(F\) be the fixed field of \(A_ n\subset G(K/k)\). Let \({\mathfrak O}_ k=\mathbb{Z}(t_ 1,\ldots,t_ n)\) and \({\mathfrak O}_ K\) (resp. \({\mathfrak O}_ F)\) be the integral closure of \({\mathfrak O}_ k\) in \(K\) (resp. \(F)\). Let \(\varphi:\text{Spec} {\mathfrak O}_ k\to\text{Spec} {\mathfrak O}_ F\) be the natural morphism. The author proves that \(\varphi\) is étale at all points of codimension 1. The author considers a quantitative version of Yamamoto’s result on specializations. The paper is very instructive, has two conjectures and in appendix 1 a nice property of the resultant of two polynomials over an integral domain \(A\).

MSC:

14A05 Relevant commutative algebra
12F10 Separable extensions, Galois theory
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
12F05 Algebraic field extensions
11R09 Polynomials (irreducibility, etc.)
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