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 O''X''. 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 b_i 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

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