Gauss's theorem (UFD ⇒ polynomial ring is UFD)
If R is a UFD, then the polynomial ring R[x] is again a UFD (and likewise in finitely many variables).
Gauss's theorem: If is a UFD, then the polynomial ring is a UFD. More generally, is a UFD for all .
A unique factorization domain (UFD) is an integral domain such that:
- Every nonzero nonunit element can be written as a finite product of irreducible elements, and
- This factorization is unique up to reordering and replacing factors by associates.
Let be a commutative ring with . The polynomial ring consists of finite sums with coefficients , with addition termwise and multiplication determined by distributivity and .