Okutsu-Montes representations of prime ideals of one-dimensional integral closures
Let \(K\) be a number field generated by a root of a monic irreducible polynomial \(f\) with integral coefficients. \textit{Ø. Ore} [Acta Math. 44, 219--314 (1923; JFM 49.0698.04); 5. Skand. Math.-Kongress. Helsingfors 1922, S. 177--180 (1923; JFM 49.0698.03)] gave a method for computing the prime ideal decomposition of a prime number \(p\) if \(f\) satisfies some regularity conditions. \textit{S. MacLane} [Duke Math. J. 2, 492--510 (1936; Zbl 0015.05801, JFM 62.0096.02); Trans. Am. math. Soc. 40, 363--395 (1936; Zbl 0015.29202, JFM 62.1106.02)] took care of these exceptions, and \textit{J. Montes} [Newton polygons of higher order and arithmetic applications. (Spanish) Ph. D. thesis, Barcelona (1999)] developed an algorithm based on these ideas. In the current article, the author gives a survey of the ideas behind Montes' algorithm and explains the relevant notions, from Ore's work up to the representation of prime ideals in one-dimensional integral closures due to \textit{K. Okutsu} [Proc. Japan Acad., Ser. A 58, 47--49 (1982; Zbl 0522.13004); ibid. 87--89 (1982; Zbl 0526.13008); ibid. 117--119 (1982; Zbl 0522.13005); ibid. 167--169 (1982; Zbl 0522.13006)] and Montes.
scientific article; zbMATH DE number 5931140