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Integer polynomials with roots mod \(p\) for all primes \(p\). (English) Zbl 1052.12002
Summary: Let \(f(X)\) be an integer polynomial which is a product of two irreducible factors. Assume that \(f(X)\) has a root mod \(p\) for all primes \(p\). If the splitting field of \(f(X)\) over the rationals is a cyclic extension of the stem fields, then the Galois group of \(f(X)\) over the rationals is soluble and of bounded Fitting length. Moreover, the fixed groups of the stem extensions are in, some sense, unique.

12F10 Separable extensions, Galois theory
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
Full Text: DOI
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