Computing Galois groups by means of Newton polygons. (English) Zbl 1071.11066

Summary: Newton polygons are useful for computing decompositions of primes in extension rings, and for computing Galois groups [see e.g. A. Movahhedi and A. Salinier, J. Lond. Math. Soc., II. Ser. 53, No. 3, 433–440 (1996; Zbl 0862.11063), A. Hermez and A. Salinier, J. Number Theory 90, No. 1, 113–129 (2001; Zbl 0990.12004), and B. Plans and N. Vila, J. Algebra 266, No. 1, 27–33 (2003; Zbl 1057.12003)]. Suppose \(f\) is a polynomial with coefficients in an algebraic number field \(K\) and \({\mathfrak p}\) is a finite prime of \(K\). Then, following Ö. Ore [Math. Ann. 99, 84–117 (1928; JFM 54.0191.02)], one can associate to \(f\) certain polynomials \(f_m\in K[X]\) according to the slopes, \(m\), of the sides of its Newton polygon with respect to \({\mathfrak p}\). Under some mild assumptions the Galois groups of the \(f_m\), viewed as polynomials over the \({\mathfrak p}\)-adic completion \(K_{\mathfrak p}\), turn out to be constituents of \(\text{Gal}_{K_{\mathfrak p}}(f)\). The point is that the \(f_m\) are usually much easier to handle than \(f\), often they are pure polynomials.


11S20 Galois theory
11R32 Galois theory
12F10 Separable extensions, Galois theory
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