Explicit construction of integral bases of radical function fields

Qingquan Wu

doi:10.5802/jtnb.714

Qingquan Wu ¹

¹ Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary, Alberta T2N 1N4

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 259-270.

Résumé
Abstract

Nous donnons une construction explicite d’une base entière pour le corps de fonction $K = k (t, ρ)$ , où $ρ^{n} = D \in k [t]$ , sous l’hypothèse $[K : k (t)] = n$ et $c h a r (k) ∤ n$ . Le discriminant du corps $K$ est également calculé. Nous expliquons pourquoi ces questions sont considérablement plus faciles que dans le cas des corps de nombres. Quelques formules pour les $P$ -signatures des corps de fonction radiciels sont également présentées dans ce papier.

We give an explicit construction of an integral basis for a radical function field $K = k (t, ρ)$ , where $ρ^{n} = D \in k [t]$ , under the assumptions $[K : k (t)] = n$ and $c h a r (k) ∤ n$ . The field discriminant of $K$ is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the $P$ -signatures of a radical function field are also discussed in this paper.

MR Zbl

DOI : 10.5802/jtnb.714

Affiliations des auteurs :

Qingquan Wu ¹

¹ Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary, Alberta T2N 1N4

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     title = {Explicit construction of integral bases of radical function fields},
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     pages = {259--270},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
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Qingquan Wu. Explicit construction of integral bases of radical function fields. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 259-270. doi : 10.5802/jtnb.714. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.714/

Bibliographie
Cité par

[1] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language. J. Symb. Comp. 24 (1997), 235–265. | MR | Zbl

[2] J. A. Buchmann and H. W. Lenstra Jr., Approximating rings of integers in number fields. J. Theor. Nombres Bordeaux 6 (1994), 221–260. | Numdam | MR | Zbl

[3] H. Cohen, A course in computational algebraic number theory. Springer-Verlag, 1993. | MR | Zbl

[4] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen. Akademische Verlagsgesellschaft, 1954. | MR | Zbl

[5] J. G. Huard, B. K. Spearman, and K. S. Williams, Integral bases for quartic fields with quadratic subfields. J. Number Theory 51 (1995), 103–117. | MR | Zbl

[6] R. H. Hudson and K. S. Williams, The integers of a cyclic quartic field. Rocky Mountain J. Math. 20 (1990), 145–150. | MR | Zbl

[7] T. W. Hungerford, Algebra. Springer-Verlag, 1974. | MR | Zbl

[8] K. F. Ireland and M. Rosen, A classical introduction to modern number theory. Springer-Verlag, 1990. | MR | Zbl

[9] KANT/KASH, Computational Algebraic Number Theory/KAnt SHell. http://www.math.tu-berlin.de/ $\sim$ kant/kash.

[10] E. Lamprecht, Verzweigungsordnungen, Differenten und Ganzheitsbasen bei Radikalerweiterungen. I. Arch. Math. 56 (1991), 569–585. | MR | Zbl

[11] E. Lamprecht, Existence of and computation of integral bases. Acta Math. Inform. Univ. Ostrav. 6 (1998), 121–128. | MR | Zbl

[12] H. B. Mann, On integral basis. Proc. Amer. Math. Soc. 9 (1958), 167–172. | MR | Zbl

[13] H. B. Mann and W. Y. Vélez, Prime ideal decomposition in $F (\sqrt[m]{μ})$ . Monatsh. Math. 81 (1976), 131–139. | MR | Zbl

[14] D. Marcus, Number Fields. Springer-Verlag, 1977. | MR | Zbl

[15] L. R. McCulloh, Integral bases in Kummer extensions of Dedekind fields. Canad. J. Math. 15 (1963), 755–765. | MR | Zbl

[16] M. J. Norris and W. Y. Vélez, Structure theorems for radical extensions of fields. Acta Arith. 38 (1980/81), 111–115. | MR | Zbl

[17] K. Okutsu, Integral basis of the field $ℚ (\sqrt[n]{a})$ . Proc. of Japan Acad. Ser. A Math. Sci. 58 (1982), 219–222. | MR | Zbl

[18] M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory. Cambridge University Press, 1997. | MR | Zbl

[19] P. Ribenboim, Algebraic numbers. John-Wiley & Sons. Inc., 1972. | MR | Zbl

[20] M. Rosen, Number theory in function fields. Springer-Verlag, 2002. | MR | Zbl

[21] R. Scheidler and A. Stein, Voronoi’s algorithm in purely cubic congruence function fields of unit rank 1. Math. Comp. 69 (2000), 1245–1266. | MR | Zbl

[22] H. Stichtenoth, Algebraic function fields and codes. Springer-Verlag, 1993. | MR | Zbl

[23] M. van Hoeij, An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comp. 18 (1994), 353–363. | MR | Zbl

[24] W. Y. Vélez, Prime ideal decomposition in $F (μ^{1 / p})$ . Pacific J. Math. 75 (1978), 589–600. | MR | Zbl

[25] P. G. Walsh, A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function. Math. Comp. 69 (2000), 1167–1182. | MR | Zbl

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