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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.

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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.

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    $\begingroup$ Is it correct to say that what you call a “Hahn embedding” in the paper is a highly nonconstructive object? If so, are there any results attempting to quantify how many such distinct representations of an ordered field $K$ exist? $\endgroup$
    – Monroe Eskew
    Commented Jan 11, 2021 at 15:38
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    $\begingroup$ @MonroeEskew Yes, as with Hahn's classical embedding theorem for ordered Abelian groups, one makes use of the axiom of choice to prove the analogous embedding theorem for ordered fields. As I mentioned in an earlier post (mathoverflow.net/questions/128935/…), it is an open question if Hahn's theorem is equivalent to the Axiom of Choice. The same question can also be raised for the embedding theorem for ordered fields. $\endgroup$
    – Philip Ehrlich
    Commented Jan 12, 2021 at 14:52
  • $\begingroup$ Is there any canonical notion of a decomposition of a field element into “components” in various Archimedean classes? Obviously there is the Archimedean class of the element itself, but what about in smaller classes? The representation in a Hahn field gives a presentation as a vector, but can we recover those coordinates in some natural way as we can in Hilbert spaces? $\endgroup$
    – Monroe Eskew
    Commented Jan 12, 2021 at 16:02
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    $\begingroup$ There is a canonical decomposition for surreal numbers--the normal forms, or Conway names as I call them--but that makes use of the simplicity hierarchy. Without the simplicity hierarchy, the decomposition depends on various selections. For the surreal case, see e.g. my "Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers", J Sym Log 66 (2001), pp. 1231-1258. And for the classical case, see e.g. Mourgues and Ressayre "Every real closed field has an integer part", J. Sym Log 58 (1993), pp. 641-647. $\endgroup$
    – Philip Ehrlich
    Commented Jan 13, 2021 at 15:49

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