Weierstrass preparation theorem

In algebra, the Weierstrass preparation theorem described a canonical form for formal power series over a complete local ring.

Let O be a complete local ring and f a formal power series in OX. Then f can be written uniquely in the form


 * $$f = (X^n + b_{n-1}X^{n-1} + \cdots + b_0) u(x), \,$$

where the bi are in the maximal ideal m of O and u is a unit of OX.

The integer n defined by the theorem is the Weierstrass degree of f.