Formal power series

Formal power series in mathematics are an analog of the power series of analysis.

Definition

For a commutative ring A with unit element denote the ring of the consequences

With componentwise addition and convolution as multiplication,

The elements of are called formal power series and are called

Written, wherein

(Compare polynomial )

Properties

• The units of are precisely those power series whose constant term is a unit in A.
• If A is a Noetherian ring or an integral domain, so this is true for each.
• If k is a field, then k [ [X ] ] is a complete discrete valuation ring. It is the completion of the polynomial ring k [ X] with respect to the ideal ( X). Its residue field k be the quotient field of the body of formal Laurent series k ( (X)).
• Conversely, according to the structure of sets of Irving S. Cohen each complete discrete valuation ring of equal characteristic is isomorphic to the ring of formal power series over its residue field.