This theorem, which extends Hahn's embedding theorem for ordered abelian groups to ordered fields, has a complicated history that makes it difficult to attribute it to any single author. However, by the early 1950s, as a result of the work of Kaplansky (1942), it appears to have assumed the status of a ``folk theorem'' among knowledgeable field theorists, with numerous proofs published thereafter. For an in-depth history of the embedding theorem along with numerous references to proofs, see my paper:



*Hahn’s Über die Nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them*. In:**From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics**, edited by J. Hintikka, Kluwer Academic Publishers, Dordrecht, 1995, pp. 165-213.


https://www.researchgate.net/publication/325019391_Hahn's_Uber

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Strictly speaking, the theorem I am referring to is much deeper than the embedding theorem stated in the question, though I suspect it is the one that is intended. A weak formulation of the ordered-field-theoretic generalization of Hahn's Embedding Theorem asserts: If $K$ is an ordered field and $\Gamma$ is its ordered Abelian group of Archimedean classes, then there is an embedding of $K$ into the Hahn field $\mathbb{R}[[X^\Gamma ]]$. For increasingly stronger versions of the theorem, see the paper cited above.