On some subgroups of the multiplicative group of finite rings

José Felipe Voloch[1]

  • [1] Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 233-239
  • ISSN: 1246-7405

Abstract

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Let S be a subset of F q , the field of q elements and h ∈ F q [ x ] a polynomial of degree d > 1 with no roots in S . Consider the group generated by the image of { x - s ∣ s ∈ S } in the group of units of the ring F q [ x ] / ( h ) . In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.

How to cite

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Voloch, José Felipe. "On some subgroups of the multiplicative group of finite rings." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 233-239. <http://eudml.org/doc/249252>.

@article{Voloch2004,
abstract = {Let $S$ be a subset of $\{\mathbf\{F\}\}_q$, the field of $q$ elements and $h \in \{\mathbf\{F\}\}_q[x]$ a polynomial of degree $d&gt;1$ with no roots in $S$. Consider the group generated by the image of $\lbrace x-s \mid s \in S\rbrace $ in the group of units of the ring $\{\mathbf\{F\}\}_q[x]/(h)$. In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.},
affiliation = {Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA},
author = {Voloch, José Felipe},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {polynomial rings; linear polynomials; primality testing},
language = {eng},
number = {1},
pages = {233-239},
publisher = {Université Bordeaux 1},
title = {On some subgroups of the multiplicative group of finite rings},
url = {http://eudml.org/doc/249252},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Voloch, José Felipe
TI - On some subgroups of the multiplicative group of finite rings
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 233
EP - 239
AB - Let $S$ be a subset of ${\mathbf{F}}_q$, the field of $q$ elements and $h \in {\mathbf{F}}_q[x]$ a polynomial of degree $d&gt;1$ with no roots in $S$. Consider the group generated by the image of $\lbrace x-s \mid s \in S\rbrace $ in the group of units of the ring ${\mathbf{F}}_q[x]/(h)$. In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.
LA - eng
KW - polynomial rings; linear polynomials; primality testing
UR - http://eudml.org/doc/249252
ER -

References

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  1. M. Agrawal, N. Kayal, N. Saxena, PRIMES is in P. http://www.cse.iitk.ac.in/news/primality.html. Zbl1071.11070
  2. D. Bernstein, Proving primality after Agrawal-Kayal-Saxena. http://cr.yp.to/papers.html. 
  3. D.  Bernstein, Sharper ABC-based bounds for congruent polynomials. http://cr.yp.to/ntheory.html. Zbl1093.11019
  4. F. Chung, Diameters and Eigenvalues. JAMS 2 (1989), 187–196. Zbl0678.05037MR965008
  5. S.D. Cohen, Polynomial factorisation and an application to regular directed graphs. Finite Fields and Appl. 4 (1998), 316–346. Zbl0957.11049MR1648561
  6. G. Frey, M. Perret, H. Stichtenoth, On the different of abelian extensions of global fields. Coding theory and algebraic geometry (Luminy, 1991), Lecture Notes in Math. 1518, 26–32. Springer, Berlin, 1992. Zbl0776.11067MR1186413
  7. N.M. Katz, Factoring polynomials in finite fields: An application of Lang-Weil to a problem of graph theory. Math. Annalen 286 (1990), 625–637. Zbl0708.05029MR1045392
  8. H.W. Lenstra Jr., Primality testing with cyclotomic rings. 
  9. W.F. Lunnon, P.A.B. Pleasants, N.M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I. Acta Arith. 35 (1979), 1–16. Zbl0408.10006MR536875
  10. I. Shparlinski, The number of different prime divisors of recurrent sequences. Mat. Zametki 42 (1987), 494–507. Zbl0635.10006MR917803
  11. J.F. Voloch, Jacobians of curves over finite fields. Rocky Mountain Journal of Math. 30 (2000), 755–759. Zbl1016.11023MR1787011

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