Puiseux Series - Field of Puiseux Series

Field of Puiseux Series

If K is a field then we can define the field of Puiseux series with coefficients in K (or over K) informally as the set of formal expressions of the form

where n and are a nonzero natural number and an integer respectively (which are part of the datum of f): in other words, Puiseux series differ from formal Laurent series in that they allow for fractional exponents of the indeterminate as long as these fractional exponents have bounded denominator (here n), and just as Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded (here by ). Addition and multiplication are as expected: one might define them by first “upgrading” the denominator of the exponents to some common denominator and then performing the operation in the corresponding field of formal Laurent series.

In other words, the field of Puiseux series with coefficients in K is the union of the fields (where n ranges over the nonzero natural numbers), where each element of the union is a field of formal Laurent series over (considered as an indeterminate), and where each such field is considered as a subfield of the ones with larger n by rewriting the fractional exponents to use a larger denominator (e.g., is identified with as expected).

This yields a formal definition of the field of Puiseux series: it is the direct limit of the direct system, indexed over the non-zero natural numbers n ordered by divisibility, whose objects are all (the field of formal Laurent series, which we rewrite as

for clarity),

with a morphism

being given, whenever m divides n, by .