Weierstrass preparation theorem

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In algebra, the Weierstrass preparation theorem describes 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 O''X''. Then f can be written uniquely in the form

f = ( X n + b n − 1 X n − 1 + ⋯ + b 0 ) u ( x ) , {\displaystyle 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 O''X''.

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

## References[edit]

* Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 208-209. ISBN 0-201-55540-9.