In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907 (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group (typically Q {\displaystyle \mathbb {Q} } or R {\displaystyle \mathbb {R} } ). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem.
Formulation The field of Hahn series K [ [ T Γ ] ] {\displaystyle K\left[\left[T^{\Gamma }\right]\right]} (in the indeterminate T {\displaystyle T} ) over a field K {\displaystyle K} and with value group Γ {\displaystyle \Gamma } (an ordered group) is the set of formal expressions of the form
f = ∑ e ∈ Γ c e T e {\displaystyle f=\sum _{e\in \Gamma }c_{e}T^{e}}
with c e ∈ K {\displaystyle c_{e}\in K} such that the support supp f := { e ∈ Γ : c e ≠ 0 } {\displaystyle \operatorname {supp} f:=\{e\in \Gamma :c_{e}\neq 0\}} of f is well-ordered. The sum and product of
f = ∑ e ∈ Γ c e T e {\displaystyle f=\sum _{e\in \Gamma }c_{e}T^{e}} and g = ∑ e ∈ Γ d e T e {\displaystyle g=\sum _{e\in \Gamma }d_{e}T^{e}}
are given by
f + g = ∑ e ∈ Γ ( c e + d e ) T e {\displaystyle f+g=\sum _{e\in \Gamma }(c_{e}+d_{e})T^{e}}
and
f g = ∑ e ∈ Γ ( ∑ e ′ + e ″ = e c e ′ d e ″ ) T e {\displaystyle fg=\sum _{e\in \Gamma }\left(\sum _{e'+e''=e}c_{e'}d_{e''}\right)T^{e}}
(in the latter, the sum ∑ e ′ + e ″ = e {\displaystyle \sum _{e'+e''=e}} over values ( e ′ , e ″ ) {\displaystyle (e',e'')} such that c e ′ ≠ 0 {\displaystyle c_{e'}\neq 0} , d e ″ ≠ 0 {\displaystyle d_{e''}\neq 0} and e ′ + e ″ = e {\displaystyle e'+e''=e} is finite because a well-ordered set cannot contain an infinite decreasing sequence). For example, T − 1 / p + T − 1 / p 2 + T − 1 / p 3 + ⋯ {\displaystyle T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots } is a Hahn series (over any field) because the set of rationals
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