## 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.

### MSC:

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

### Citations:

JFM 54.0191.02; Zbl 0862.11063; Zbl 0990.12004; Zbl 1057.12003
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