The Universality of Formal Power Series Fields*

In a recent paper,f André Gleyzal has constructed ordered fields consisting of certain "transfinite real numbers" and has established the interesting result that any ordered field can be considered as a subfield of one of these transfinite fields. These fields prove to be identical with fields of formal power series in which the exponents are allowed to range over a suitable ordered abelian group. Such fields were first introduced by Hahn,$ while they have been analyzed in terms of generalized valuations by Krull.§ Gleyzal applied his construction of transfinite real numbers not only to the case when the coefficient field consisted of real numbers, but also to suitable fields of characteristic p. He conjectured that this construction should yield a "universal" field of characteristic p. We show here that KrmTs technique can be used to establish Gleyzal's conjecture.