On some subfields of K((X))

Shaheen Nazir
School of Mathematical Science, GC University, Lahore, 68-B, New Muslim Town, Lahore, Pakistan.
Angel Popescu
Technical University of Civil Engineering, Bucharest, Romania.

\(^{1}\)Corresponding Author: shaheen.nazir@gmail.com

Copyright © 2006 Shaheen Nazir, Angel Popescu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: December, 2006.

Abstract

Let K be a commutative field and let \(K((X))\) be the field of Laurent series in one variable \(X\), consider with its natural \(X-\)adic topology. In this paper we prove that any closed subfield \(K ⊂ L ⊂ K((X))\) is of the form \(L = K((f))\) and \(K((X))\) is algebraic over L of degree \(ord_{X}(f)\). Some other properties of \(L\) are studied.

Keywords:

Field of Laurent series, Krull valuation, algebraic extension, completeness.