Explicit construction of integral bases of radical function fields

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 (2010)

  • Volume: 22, Issue: 1, page 259-270
  • ISSN: 1246-7405

Abstract

top
We give an explicit construction of an integral basis for a radical function field K = k ( t , ρ ) , where ρ n = D ∈ 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.

How to cite

top

Wu, Qingquan. "Explicit construction of integral bases of radical function fields." Journal de Théorie des Nombres de Bordeaux 22.1 (2010): 259-270. <http://eudml.org/doc/116399>.

@article{Wu2010,
abstract = {We give an explicit construction of an integral basis for a radical function field $K=k(t,\rho )$, where $\rho ^n=D\in k[t]$, under the assumptions $[K:k(t)]=n$ and $\mbox \{char\}(k)\nmid n$. 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.},
affiliation = {Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary, Alberta T2N 1N4},
author = {Wu, Qingquan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {integral bases; radical function fields},
language = {eng},
number = {1},
pages = {259-270},
publisher = {Université Bordeaux 1},
title = {Explicit construction of integral bases of radical function fields},
url = {http://eudml.org/doc/116399},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Wu, Qingquan
TI - Explicit construction of integral bases of radical function fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 259
EP - 270
AB - We give an explicit construction of an integral basis for a radical function field $K=k(t,\rho )$, where $\rho ^n=D\in k[t]$, under the assumptions $[K:k(t)]=n$ and $\mbox {char}(k)\nmid n$. 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.
LA - eng
KW - integral bases; radical function fields
UR - http://eudml.org/doc/116399
ER -

References

top
  1. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language. J. Symb. Comp. 24 (1997), 235–265. Zbl0898.68039MR1484478
  2. J. A. Buchmann and H. W. Lenstra Jr., Approximating rings of integers in number fields. J. Theor. Nombres Bordeaux 6 (1994), 221–260. Zbl0828.11075MR1360644
  3. H. Cohen, A course in computational algebraic number theory. Springer-Verlag, 1993. Zbl0786.11071MR1228206
  4. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen. Akademische Verlagsgesellschaft, 1954. Zbl0057.27301MR66417
  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. Zbl0826.11048MR1321725
  6. R. H. Hudson and K. S. Williams, The integers of a cyclic quartic field. Rocky Mountain J. Math. 20 (1990), 145–150. Zbl0707.11078MR1057983
  7. T. W. Hungerford, Algebra. Springer-Verlag, 1974. Zbl0293.12001MR600654
  8. K. F. Ireland and M. Rosen, A classical introduction to modern number theory. Springer-Verlag, 1990. Zbl0712.11001MR1070716
  9. KANT/KASH, Computational Algebraic Number Theory/KAnt SHell. http://www.math.tu-berlin.de/ ∼ kant/kash. 
  10. E. Lamprecht, Verzweigungsordnungen, Differenten und Ganzheitsbasen bei Radikalerweiterungen. I. Arch. Math. 56 (1991), 569–585. Zbl0735.11062MR1106499
  11. E. Lamprecht, Existence of and computation of integral bases. Acta Math. Inform. Univ. Ostrav. 6 (1998), 121–128. Zbl1024.11074MR1822521
  12. H. B. Mann, On integral basis. Proc. Amer. Math. Soc. 9 (1958), 167–172. Zbl0081.26602MR93502
  13. H. B. Mann and W. Y. Vélez, Prime ideal decomposition in F ( μ m ) . Monatsh. Math. 81 (1976), 131–139. Zbl0324.12003MR399043
  14. D. Marcus, Number Fields. Springer-Verlag, 1977. Zbl0383.12001MR457396
  15. L. R. McCulloh, Integral bases in Kummer extensions of Dedekind fields. Canad. J. Math. 15 (1963), 755–765. Zbl0118.27503MR153668
  16. M. J. Norris and W. Y. Vélez, Structure theorems for radical extensions of fields. Acta Arith. 38 (1980/81), 111–115. Zbl0492.12011MR604227
  17. K. Okutsu, Integral basis of the field ℚ ( a n ) . Proc. of Japan Acad. Ser. A Math. Sci. 58 (1982), 219–222. Zbl0524.12002MR667638
  18. M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory. Cambridge University Press, 1997. Zbl0685.12001MR1483321
  19. P. Ribenboim, Algebraic numbers. John-Wiley & Sons. Inc., 1972. Zbl0247.12002MR340212
  20. M. Rosen, Number theory in function fields. Springer-Verlag, 2002. Zbl1043.11079MR1876657
  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. Zbl1042.11068MR1653974
  22. H. Stichtenoth, Algebraic function fields and codes. Springer-Verlag, 1993. Zbl0816.14011MR1251961
  23. M. van Hoeij, An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comp. 18 (1994), 353–363. Zbl0834.68059MR1324494
  24. W. Y. Vélez, Prime ideal decomposition in F ( μ 1 / p ) . Pacific J. Math. 75 (1978), 589–600. Zbl0344.12002MR506215
  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. Zbl1042.14039MR1710624

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.