Über das Fortsetzen von Bewertungen in vollständigen Körpern. (On the extension of valuations in complete fields) (Q1112112)

Let K be a field that is complete with respect to a valuation \(\phi\), and L a finite extension of K. Then there exists a unique valuation on L that extends \(\phi\). The usual proof of this well-known theorem uses Hensel's lemma in the non-archimedean case, and Ostrowski's classification of complete archimedean fields in the archimedean case. This proof is due to Ostrowski (1918). An earlier proof of the existence part of the theorem was given by Kürschák (1912). Instead of Ostrowski's classification theorem, which was not yet available, Kürschák employed results of Hadamard on the singularities of power series with complex coefficients. He expressed regret at the considerable technical difficulties which this entailed. In the present paper we give a direct and elementary proof of the theorem quoted above, which avoids both Hensel's lemma and Ostrowski's classification. The archimedean or non-archimedean character of the valuation does not play a role in the argument.