Rational fractions and formal series

Although polynomials are the most elementary tools of formal calculus, they are not sufficient to fully express the general operations of commutative algebra. That's why, to allow for the possibility of division, it's natural to introduce the notion of rational fraction, which is to the polynomial what the fraction (also called rational number) is to the integer. A general construction procedure is then revealed, that of the body of fractions of an integral ring.

In addition, limited developments, developments in whole series, and other developments carried out either to an arbitrary order or in an unlimited manner, require the introduction of suitable tools, which are expressed in the framework of formal series.

Like polynomials, rational fractions and formal series are particularly well-suited to formal manipulations, which can be carried out using software.