Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T02:06:02.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Form of Cassels’ p-adic Embedding Theorem for Number Fields

Published online by Cambridge University Press:  20 November 2018

Arturas Dubickas ,
Min Sha  and
Igor Shparlinski
Arturas Dubickas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, LT-03225 Vilnius, Lithuania. e-mail: [email protected]
Min Sha
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. e-mail: [email protected], [email protected]
Igor Shparlinski
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we give a general explicit form of Cassels’ $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

Keywords

11R0411S8511G5011R0911R18number fieldp-adic embeddingheightpolynomialcyclotomic field
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 5 , 01 October 2015 , pp. 1046 - 1064
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Bellaïche, J., Théorème de Chebotarev et complexité de Littlewood. arxiv:1308.1022.Google Scholar
[2] Brawley, J. W. and Gao, S., On density of primitive elements for field extensions. http://www.math.clemson.edu/~sgao/papers/prim-ele.pdf. Google Scholar
[3] Carmichael, R. D. and Mason, T. E., Note on the roots of algebraic equations. Bull. Amer. Math. Soc. 21(1914), no. 1, 14–22.http://dx.doi.org/10.1090/S0002-9904-1914-02563-7 Google Scholar
[4] Cassels, J. W. S., An embedding theorem for fields. Bull. Austral. Math. Soc. 14(1976), no. 2, 193–198.http://dx.doi.org/10.1017/S0004972700025O3X Google Scholar
[5] Everest, G., van der Poorten, A. J., Shparlinski, I. E., and Ward, T., Recurrence sequences. Mathematical Surveys and Monographs, 104, American Mathematical Society, Providence, RI, 2003.Google Scholar
[6] Klinger, A., The Vandermonde matrix. Amer. Math. Monthly, 74(1967), 571–574.http://dx.doi.org/10.2307/2314898 Google Scholar
[7] Konyagin, S. V., On the number of solutions of an nth degree congruence with one unknown. Matem. Sb. (N.S.) 109(151)(1979), no. 2,171–187; English version: Math. USSR-Sb. 37(1980), 151–166.Google Scholar
[8] Konyagin, S. V. and Steger, T., On polynomial congruences. Math. Notes 55(1994), no. 5–6, 596–600.Google Scholar
[9] Landau, E., Über eine Aufgabe der Funktionentheorie. Tohoku Math. J. 5(1914), 97–116.Google Scholar
[10] Loxton, J. H. and van der Poorten, A. J., On the growth of recurrence sequences. Math. Proc. Cambridge Philos. Soc. 81(1977), 369–376.http://dx.doi.Org/10.1017/S0305004100053445 Google Scholar
[11] Mahler, K., An inequality for the discriminant of a polynomial. Michigan Math. J. 11(1964), 257–262.http://dx.doi.org/10.1307/mmjV1028999140 Google Scholar
[12] Maynard, J., Small gaps between primes. Annals of Math. 181(2015), no. 1, 383–413.http://dx.doi.Org/10.4007/annals.2015.181.1.7 Google Scholar
[13] Odlyzko, A. M., Some analytic estimates of class numbers and discriminants. Invent. Math. 29(1975), no. 3, 275–286.http://dx.doi.Org/10.1007/BF01389854 Google Scholar
[14] van der Poorten, A. J. and Schlickewei, H.-P., Additive relations infields. J. Austral. Math. Soc. 51(1991), no. 1, 154–170. http://dx.doi.Org/10.1017/S144678870003336X Google Scholar
[15] van der Poorten, A. J. and Shparlinski, I. E., On the number of zeros of exponential polynomials and related questions. Bull. Austral. Math. Soc. 46(1992), no. 3, 401–412.http://dx.doi.Org/10.1017/S0004972700012065 Google Scholar
[16] van der Poorten, A. J. and Shparlinski, I. E., On sequences of polynomials defined by certain recurrence relations. Acta Sci. Math. (Szeged) 61(1995), no. 1–4, 77–103.Google Scholar
[17] Specht, W., Abschätzungen der Wurzeln algebraischer Gleichungen. Math. Z. 52(1949), 310–321.http://dx.doi.org/10.1007/BF02230697 Google Scholar
[18] Stewart, C. L., On the number of solutions of polynomial congruences and Thue equations. J. Amer. Math. Soc. 4(1991), no. 4, 793–835. http://dx.doi.Org/10.1090/S0894-0347-1991-1119199-X Google Scholar
[19] Vaaler, J. D. and Widmer, M., A note on small generators of number fields. In: Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics, 587, American Mathematical Society, Providence, RI, 2013.Google Scholar
[20] Waldschmidt, M., Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables. Grundlehren der Mathematischen Wissenschaften, 326, Springer, Berlin, 2000.Google Scholar
[21] Washington, L. C., Introduction to cyclotomic fields. Graduate Texts in Mathematics, 83, Springer, New York, 1982.Google Scholar
[22] Widmer, M., Counting primitive points of bounded height. Trans. Amer. Math. Soc. 362(2010), no. 9, 4793–4829.http://dx.doi.org/10.1090/S0002-9947-10-05173-1 Google Scholar
[23] Xylouris, T., On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet L-functions. Acta Arith. 150(2011), no. 1, 65–91.http://dx.doi.Org/10.4064/aa150-1-4 Google Scholar
[24] Zhang, Y., Bounded gaps between primes. Ann. of Math. 179(2014), no. 3, 1121–1174.http://dx.doi.Org/10.4007/annals.2014.179.3.7 Google Scholar