# Prove that $K[x_1, \ldots]$ is a UFD [duplicate]

Possible Duplicate:
An example of a non Noetherian UFD

How can I prove that $K[x_1, \ldots]$, with $K$ a field, is a UFD? That means there's a unique factorization. But how to prove it?

## marked as duplicate by anon, Clayton, JSchlather, Zev ChonolesJan 21 '13 at 19:11

• Can you prove that $K[x_1,\ldots,x_n]$ is a UFD? Every element lives in such a subring, and you can use arguments about the degree of other variables to prove that the factorization in this ring is the only factorization in the larger ring. – Brett Frankel Jan 21 '13 at 18:50