There is a theorem attributed to Hahn that every ordered field $F$ containing $\mathbb R$ is a subfield of a formal power series field $\mathbb R[[X^\Gamma]]$, where $\Gamma$ is an ordered abelian group. Can you give a nice reference in English for a proof of this theorem? Or if it is not too hard, please sketch a proof below.



Hahn’s theorem on ordered fields
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](https://www.researchgate.net/publication/325019391_Hahn%27s_Uber)


Edit:


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.


