Hahn Series - Formulation

Formulation

The field of Hahn series (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 such that the support of f is well-ordered. The sum and product of and are given by and (in the latter, the sum over values such that and is finite because a well-ordered set cannot contain an infinite decreasing sequence).

For example, 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 so it is algebraic over .)

The valuation of is defined as the smallest e such that (in other words, the smallest element of the support of f): this makes into a spherically complete valued field with value group Γ (justifying a posteriori the terminology); in particular, v defines a topology on . If, then v corresponds to an ultrametric) absolute value, with respect to which 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 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").

If K is algebraically closed (but not necessarily of characteristic zero) and Γ is divisible, then is algebraically closed. Thus, the algebraic closure of is contained in (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 in positive characteristic as a subset of .

If K is an ordered field then 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 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 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 : e.g., it is algebraically closed or real closed when K is so and Γ is divisible.