Geodesic
An algebraically closed field
Glasgow mathematical journal, Tome 9 (1968) no. 2, pp. 146-151

Voir la notice de l'article provenant de la source Cambridge University Press

Let k be any algebraically closed field, and denote by k((t)) the field of formal power series in one indeterminate t over k. Letso 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. 
Rayner, F. J. An algebraically closed field. Glasgow mathematical journal, Tome 9 (1968) no. 2, pp. 146-151. doi: 10.1017/S0017089500000422
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