They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907[1] (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
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.
The field of Hahn series
(an ordered group) is the set of formal expressions of the form with
is finite because a well-ordered set cannot contain an infinite decreasing sequence).
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
of a non-zero Hahn series is defined as the smallest
(in other words, the smallest element of the support of
is up to (non-unique) isomorphism the only spherically complete valued field with residue field
corresponds to an ultrametric absolute value
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
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"[4]).
is algebraically closed (but not necessarily of characteristic zero) and
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
is totally ordered by making the indeterminate
infinitesimal (greater than 0 but less than any positive element of
) or, equivalently, by using the lexicographic order on the coefficients of the series.
[7] 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[8]).
If κ is an infinite regular cardinal, one can consider the subset of
consisting of series whose support set
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
[9] One can define a notion of summable families in
is a family of Hahn series
, and we have[10] and This notion of summable family does not correspond to the notion of convergence in the valuation topology on
denote the ring of real-valued functions which are analytic on a neighborhood of
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:[12] 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
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