Weierstrass preparation theorem: Difference between revisions
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In [[algebra]], the '''Weierstrass preparation theorem''' | {{subpages}} | ||
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 | 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 | ||
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==References== | ==References== | ||
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=208-209 }} | * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=208-209 }} | ||
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Latest revision as of 07:01, 7 November 2024
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
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
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 208-209. ISBN 0-201-55540-9.