Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T14:57:03.709Z Has data issue: false hasContentIssue false
  • English
  • Français

On p-adic analytic continuation with applications to generating elements

Part of: Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  10 June 2015

Victor Alexandru ,
Marian Vâjâitu  and
Alexandru Zaharescu
Victor Alexandru
Affiliation:
Department of Mathematics, University of Bucharest, 14 Academiei Street, 010014 Bucharest, Romania
Marian Vâjâitu
Affiliation:
Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 5, PO Box 1-764, 014700 Bucharest, Romania ([email protected])
Alexandru Zaharescu
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a prime number p and the Galois orbit O(T) of an integral transcendental element T of , the topological completion of the algebraic closure of the field of p-adic numbers, we study the p-adic analytic continuation around O(T) of functions defined by limits of sequences of restricted power series with p-adic integer coefficients. We also investigate applications to generating elements for or for some classes of closed subfields of .

Keywords

p-adic analytic functionslocal fieldsgenerating elements

MSC classification

Secondary: 11S99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 1 - 10
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Alexandru, V. and Zaharescu, A., A transcendence criterion over p-adic fields, Nihonkai Math. J. 14 (2003), 83–93.Google Scholar
2. Alexandru, V., Popescu, N. and Zaharescu, A., On the closed subfields of , J. Number Theory 68(2) (1998), 131–150.CrossRefGoogle Scholar
3. Alexandru, V., Popescu, N. and Zaharescu, A., Trace on , J. Number Theory 88(1) (2001), 13–48.Google Scholar
4. Alexandru, V., Popescu, N. and Zaharescu, A., The generating degree of , Can. Math. Bull. 44(1) (2001), 3–11.Google Scholar
5. Alexandru, V., Popescu, N. and Zaharescu, A., Metric invariants in associated to differential operators, Rev. Roumaine Math. Pures Appl. 46(5) (2001), 551–564.Google Scholar
6. Amice, Y., Les nombres p-adiques, Collection SUP: le Mathématicien (in French) (Presse Universitaires de France, Paris, 1975).Google Scholar
7. Artin, E., Algebraic numbers and algebraic functions (Gordon and Breach, New York, 1967).Google Scholar
8. Ax, J., Zeros of polynomials over local fields: the Galois action, J. Alg. 15 (1970), 417–428.CrossRefGoogle Scholar
9. Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean analysis (Springer, 1984).CrossRefGoogle Scholar
10. Fresnel, J. and van der Put, M., Rigid analytic geometry and its applications (Birkhäuser, 2004).Google Scholar
11. Iovita, A. and Zaharescu, A., Completions of r.a.t.-valued fields of rational functions, J. Number Theory 50(2) (1995), 202–205.Google Scholar
12. Iovita, A. and Zaharescu, A., Generating elements for , J. Math. Kyoto Univ. 39(2) (1999), 233–248.Google Scholar
13. Jacobson, N., Basic algebra II, 2nd edn (W. H. Freeman and Co., New York, 1989).Google Scholar
14. Lang, S., Algebra, 3rd edn (Springer, 2002).Google Scholar
15. Okutsu, K., Construction of integral basis, I, Proc. Jpn Acad. A58 (1982), 47–49.Google Scholar
16. Okutsu, K., Construction of integral basis, II, Proc. Jpn Acad. A58 (1982), 87–89.Google Scholar
17. Okutsu, K., Construction of integral basis, III, Proc. Jpn Acad. A58 (1982), 117–119.Google Scholar
18. Okutsu, K., Construction of integral basis, IV, Proc. Jpn Acad. A58 (1982), 167–169.Google Scholar
19. Okutsu, K., Integral basis of the field , Proc. Jpn Acad. A 58 (1982), 219–222.Google Scholar
20. Ota, K., On saturated distinguished chains over a local field, J. Number Theory 79 (1999), 217–248.Google Scholar
21. Ota, K., On saturated distinguished chains over a local field, II, J. Number Theory 111 (2005), 86–143.Google Scholar
22. Ota, K., On saturated distinguished chains over a local field, III, The case of p-extensions with two proper higher ramification groups, Int. J. Math. 17(5) (2006), 493–604.Google Scholar
23. Popescu, N. and Zaharescu, A., On the structure of the irreducible polynomials over local fields, J. Number Theory 52 (1995), 98–118.Google Scholar
24. Popescu, A., Popescu, N., Vâjâitu, M. and Zaharescu, A., Chains of metric invariants over a local field, Acta Arith. 103(1) (2002), 27–40.Google Scholar
25. Popescu, A., Popescu, N. and Zaharescu, A., Metric invariants over Henselian valued fields, J. Alg. 266(1) (2003), 14–26.Google Scholar
26. Robert, A. M., A course in p-adic analysis (Springer, 2000).Google Scholar
27. Sen, S., On automorphisms of local fields, Annals Math. (2) 90 (1969), 33–46.CrossRefGoogle Scholar
28. Singh, A. P. and Khanduja, S. K., On construction of saturated distinguished chains, Mathematika 54(1–2) (2007), 59–65.CrossRefGoogle Scholar
29. Tate, J. T., p-divisible groups, in Proceedings of a conference on local fields: NUFFIC summer school held at Driebergen (the Netherlands) in 1966, pp. 158–183 (Springer, 1967).CrossRefGoogle Scholar