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Proceedings of the American Mathematical Society
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
Published electronically: April 24, 2001
MathSciNet review: 1860477
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-01-06001-4
PII: S 0002-9939(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