Editing Surreal number (section)

===Hahn series===

Alling<ref name="Alling" />{{rp|at=th. 6.55, p. 246}} also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of [[Hahn series]] with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined [[#Powers of ω and the Conway normal form|above]]). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.

This isomorphism makes the surreal numbers into a [[valued field]] where the valuation is the additive inverse of the exponent of the leading term in the Conway normal form, e.g., {{math|1=''ν''(''ω'') = −1}}. The [[valuation ring]] then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part). The reason for the sign inversion is that the exponents in the Conway normal form constitute a reverse well-ordered set, whereas Hahn series are formulated in terms of (non-reversed) well-ordered subsets of the value group.