Puiseux Series - Generalization

Generalization

The field of Puiseux series is not complete, but its completion can be easily described: it is the field of formal expressions of the form, where the support of the coefficients (that is, the set of e such that ) is the range of an increasing sequence of rational numbers that either is finite or tends to +∞. In other words, such series admit exponents of unbounded denominators, provided there are finitely many terms of exponent less than A for any given bound A. For example, is not a Puiseux series, but it is the limit of a Cauchy sequence of Puiseux series (Puiseux polynomials). However, even this completion is still not "maximally complete" in the sense that it admits non-trivial extensions which are valued fields having the same value group and residue field, hence the opportunity of completing it even more:

Hahn series are a further (larger) generalization of Puiseux series, introduced by Hans Hahn (in the course of the proof of his embedding theorem in 1907 and then studied by him in his approach to Hilbert's seventeenth problem), where instead of requiring the exponents to have bounded denominator they are required to form a well-ordered subset of the value group (usually or ). These were later further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting (they are therefore sometimes known as Hahn-Mal'cev-Neumann series). Using Hahn series, it is possible to give a description of the algebraic closure of the field of power series in positive characteristic which is somewhat analogous to the field of Puiseux series.