Hahn series

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 formula_1 or formula_2). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him as fields in his approach to Hilbert's seventeenth problem.
Formulation.
The field of Hahn series formula_3 (in the indeterminate "T") over a field "K" and with value group "Γ" (an ordered group) is the set of formal expressions of the form
with formula_5 such that the support formula_6 of "f" is well-ordered. The sum and product of
are given by
and
(in the latter, the sum formula_11 over values formula_12 such that formula_13 and formula_14 is finite because a well-ordered set cannot contain an infinite decreasing sequence).
For example, formula_15 is a Hahn series (over any field) because the set of rationals
is well-ordered; it is not a Puiseux series because the denominators in the exponents are unbounded. (And if the base field "K" has characteristic "p", then this Hahn series satisfies the equation formula_17 so it is algebraic over formula_18.)
Properties.
Properties of the valued field.
The valuation formula_19 of a non zero Hahn series
is defined as the smallest formula_21 such that formula_22 (in other words, the smallest element of the support of formula_23): this makes formula_24 into a spherically complete valued field with value group formula_25 and residue field formula_26 (justifying "a posteriori" the terminology). In fact, if formula_26 has characteristic zero, then formula_28 is up to (non unique) isomorphism the only spherically complete valued field with residue field formula_26 and value group formula_25.
The valuation formula_31 defines a topology on formula_3. If formula_33, then "v" corresponds to an ultrametric absolute value formula_34, with respect to which formula_3 is a complete metric space. However, unlike in the case of formal Laurent series or Puiseux series, the formal sums used in defining the elements of the field do "not" converge: in the case of formula_15 for example, the absolute values of the terms tend to 1 (because their valuations tend to 0), so the series is not convergent (such series are sometimes known as "pseudo-convergent").
Algebraic properties.
If "K" is algebraically closed (but not necessarily of characteristic zero) and "Γ" is divisible, then formula_3 is algebraically closed. Thus, the algebraic closure of formula_38 is contained in formula_39, where formula_40 is the algebraic closure of formula_26 (when "K" is of characteristic zero, it is exactly the field of Puiseux series): in fact, it is possible to give a somewhat analogous description of the algebraic closure of formula_38 in positive characteristic as a subset of formula_3.
If "K" is an ordered field then formula_3 is totally ordered by making the indeterminate "T" infinitesimal (greater than 0 but less than any positive element of "K") or, equivalently, by using the lexicographic order on the coefficients of the series. If "K" is real-closed and "Γ" is divisible then formula_3 is itself real closed. This fact can be used to analyse (or even construct) the field of surreal numbers (which is isomorphic, as an ordered field, to the field of Hahn series with real coefficients and value group the surreal numbers themselves).
If "κ" is an infinite regular cardinal, one can consider the subset of formula_3 consisting of series whose support set formula_47 has cardinality (strictly) less than "κ": it turns out that this is also a field, with much the same algebraic closedness properties as the full formula_3: e.g., it is algebraically closed or real closed when "K" is so and "Γ" is divisible.
Summable families.
Summable families.
One can define a notion of summable families in formula_3. If formula_50 is a set and formula_51 is a family of Hahn series formula_52, then we say that formula_51 is summable if the set formula_54 is well-ordered, and each set formula_55 for formula_21 is finite.
We may then define the sum formula_57 as the Hahn series formula_58.
If formula_59 are summable, then so are the families formula_60, and we have
formula_61 and formula_62.
This notion of summable family does not correspond to the notion of convergence in the valuation topology on formula_3. For instance, in formula_64, the family formula_65 is summable but the sequence formula_66 does not converge.
Evaluating analytic functions.
Let formula_67< and let formula_68 denote the ring of real-valued functions which are analytic on a neighborhood of formula_69.
If formula_26 contains formula_2, then we can evaluate every element formula_23 of formula_68 at every element of formula_3 of the form formula_75, where the valuation of formula_76 is strictly positive. 
Indeed, the family formula_77 is always summable, so we can define formula_78. This defines a ring morphism formula_79.
Hahn–Witt series.
The construction of Hahn series can be combined with Witt vectors (at least over a perfect field) to form twisted Hahn series or Hahn–Witt series: for example, over a finite field "K" of characteristic "p" (or their algebraic closure), the field of Hahn–Witt series with value group "Γ" (containing the integers) would be the set of formal sums formula_80 where now formula_81 are Teichmüller representatives (of the elements of "K") which are multiplied and added in the same way as in the case of ordinary Witt vectors (which is obtained when "Γ" is the group of integers). When "Γ" is the group of rationals or reals and "K" is the algebraic closure of the finite field with "p" elements, this construction gives a (ultra)metrically complete algebraically closed field containing the "p"-adics, hence a more or less explicit description of the field formula_82 or its spherical completion.