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An algebraically closed field

Published online by Cambridge University Press:  18 May 2009

F. J. Rayner
F. J. Rayner
Affiliation:
University of Liverpool
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Let k be any algebraically closed field, and denote by k((t)) the field of formal power series in one indeterminate t over k. Let

so that K is the field of Puiseux expansions with coefficients in k (each element of K is a formal power series in tl/r for some positive integer r). It is well-known that K is algebraically closed if and only if k is of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous to K which is algebraically closed when k has non-zero characteristic p. In this paper, I prove that the set L of all formal power series of the form Σaitei (where (ei) is well-ordered, ei = mi|nprt, n ∈ Ζ, mi ∈ Ζ, ai ∈ k, ri ∈ Ν) forms an algebraically closed field.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 9 , Issue 2 , July 1968 , pp. 146 - 151
Copyright
Copyright © Glasgow Mathematical Journal Trust 1968

References

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