Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The algebraic closure of the power series field in positive characteristic


Author: Kiran S. Kedlaya
Journal: Proc. Amer. Math. Soc. 129 (2001), 3461-3470
MSC (1991): Primary 13F25; Secondary 13J05, 12J25
DOI: https://doi.org/10.1090/S0002-9939-01-06001-4
Published electronically: April 24, 2001
MathSciNet review: 1860477
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

For $K$ an algebraically closed field, let $K((t))$ denote the quotient field of the power series ring over $K$. The ``Newton-Puiseux theorem'' states that if $K$ has characteristic 0, the algebraic closure of $K((t))$is the union of the fields $K((t^{1/n}))$ over $n \in \mathbb{N} $. We answer a question of Abhyankar by constructing an algebraic closure of $K((t))$ for any field $K$ of positive characteristic explicitly in terms of certain generalized power series.


References [Enhancements On Off] (What's this?)

  • 1. S. Abhyankar, Two notes on formal power series, Proc. Amer. Math. Soc. 7 (1956), 903-905. MR 18:277a
  • 2. Ali Benhissi, La clôture algébrique du corps des séries formelles, Ann. Math. Blaise Pascal 2 (1995), no. 2, 1–14 (French). MR 1371887
  • 3. C. Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Amer. Math. Soc., 1951. MR 13:64a
  • 4. H. Hahn, Über die nichtarchimedische Größensysteme, in Gesammelte Abhandlungen I, Springer, Vienna, 1995.
  • 5. M.-F. Huang, Ph.D. thesis, Purdue University, 1968.
  • 6. K.S. Kedlaya, Power series and $p$-adic algebraic closures, xxx preprint math.AG/9906030.
  • 7. Bjorn Poonen, Maximally complete fields, Enseign. Math. (2) 39 (1993), no. 1-2, 87–106. MR 1225257
  • 8. F. J. Rayner, An algebraically closed field, Glasgow Math. J. 9 (1968), 146–151. MR 0234941, https://doi.org/10.1017/S0017089500000422
  • 9. Paulo Ribenboim, Fields: algebraically closed and others, Manuscripta Math. 75 (1992), no. 2, 115–150. MR 1160093, https://doi.org/10.1007/BF02567077
  • 10. Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • 11. Doru Ştefănescu, A method to obtain algebraic elements over 𝐾((𝑇)) in positive characteristic, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 26(74) (1982), no. 1, 77–91. MR 660937
  • 12. Doru Ştefănescu, On meromorphic formal power series, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 27(75) (1983), no. 2, 169–178. MR 710034
  • 13. Sanju Vaidya, Generalized Puiseux expansions and their Galois groups, Illinois J. Math. 41 (1997), no. 1, 129–141. MR 1433191

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13F25, 13J05, 12J25

Retrieve articles in all journals with MSC (1991): 13F25, 13J05, 12J25


Additional Information

Kiran S. Kedlaya
Affiliation: Department of Mathematics (Room 2-251), Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, 970 Evans Hall, University of California, Berkeley, California 94720
Email: kedlaya@math.mit.edu, Kedlaya@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06001-4
Keywords: Power series, generalized power series, algebraic closure, Puiseux expansions, Mal'cev-Neumann rings
Received by editor(s): November 12, 1998
Received by editor(s) in revised form: April 15, 2000
Published electronically: April 24, 2001
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2001 American Mathematical Society