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Okutsu-Montes representations of prime ideals of one-dimensional integral closures. (English) Zbl 1248.11107

Let \(K\) be a number field generated by a root of a monic irreducible polynomial \(f\) with integral coefficients. 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. 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 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 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.

MSC:

11Y40 Algebraic number theory computations
11Y05 Factorization
11R04 Algebraic numbers; rings of algebraic integers
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References:

[1] D. Ford and O. Veres, On the complexity of the Montes Ideal Factorization Algorithm, in: “Algorithmic Number Theory” , (Hanrot, G., Morain, F., and Thomé, E., eds.), 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010, Lecture Notes in Computer Science 6197 , Springer, 2010, pp. 174\Ndash185. · Zbl 1260.11085 · doi:10.1007/978-3-642-14518-6_16
[2] J. Guàrdia, J. Montes, and E. Nart, Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields, J. Théor. Nombres Bordeaux (to appear), · Zbl 1266.11131 · doi:10.5802/jtnb.782
[3] J. Guàrdia, J. Montes, and E. Nart, Higher Newton polygons and integral bases, arXiv: · Zbl 1394.11071
[4] J. Guàrdia, J. Montes, and E. Nart, Okutsu invariants and Newton polygons, Acta Arith. 145(1) (2010), 83\Ndash108. · Zbl 1266.11121 · doi:10.4064/aa145-1-5
[5] J. Guàrdia, J. Montes, and E. Nart, A new computational approach to ideal theory in number fields, · Zbl 1287.11142
[6] J. Guàrdia, J. Montes, and E. Nart, Arithmetic in big number fields: The ‘+Ideals’ package, · Zbl 1266.11131
[7] J. Guàrdia, J. Montes, and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. (to appear), arXiv: · Zbl 1252.11091 · doi:10.1090/S0002-9947-2011-05442-5
[8] J. Guàrdia, E. Nart, and S. Pauli, Single-factor lifting and factorization of polynomials over local fields, in preparation. · Zbl 1262.11106
[9] S. MacLane, A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40(3) (1936), 363\Ndash395. · Zbl 0015.29202 · doi:10.2307/1989629
[10] S. MacLane, A construction for prime ideals as absolute values of an algebraic field, Duke Math. J. 2(3) (1936), 492\Ndash510. · Zbl 0015.05801 · doi:10.1215/S0012-7094-36-00243-0
[11] J. Montes, Polígonos de Newton de orden superior y aplicaciones aritméticas, Tesi Doctoral, Universitat de Barcelona (1999).
[12] K. Okutsu, Construction of integral basis. I, Proc. Japan Acad. Ser. A Math. Sci. 58(1) (1982), 47\Ndash49; Construction of integral basis. II, Proc. Japan Acad. Ser. A Math. Sci. 58(2) (1982), 87\Ndash89. · Zbl 0526.13008 · doi:10.3792/pjaa.58.47
[13] Ø. Ore, Zur Theorie der algebraischen Körper, Acta Math. 44(1) (1923), 219\Ndash314. · JFM 49.0698.04 · doi:10.1007/BF02403925
[14] Ø. Ore, Bestimmung der Diskriminanten algebraischer Körper, Acta Math. 45(1) (1925), 303\Ndash344. · JFM 51.0142.01 · doi:10.1007/BF02395474
[15] S. Pauli, Factoring polynomials over local fields, II, in: “Algorithmic Number Theory” , (Hanrot, G., Morain, F., and Thomé, E., eds.), 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010, Lecture Notes in Computer Science 6197 , Springer, 2010, pp. 301\Ndash315. · Zbl 1260.12005 · doi:10.1007/978-3-642-14518-6_24
[16] O. Veres, On the Complexity of Polynomial Factorization Over \(P\)-adic Fields, PhD Dissertation, Concordia University (2009).
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