# Commutative ring

• Apr 8th 2010, 06:25 AM
bram kierkels
Commutative ring
Let $R$ be a commutative ring with $r \in R$.
Let $f(x)$ be an element of $R[x]$.
I want to show that $f(x)$ is irreducible over $R$ if and only if $f(x + r)$ irreducible over $R$.
• Apr 8th 2010, 07:05 AM
tonio
Quote:

Originally Posted by bram kierkels
Let $R$ be a commutative ring with $r \in R$.
Let $f(x)$ be an element of $R[x]$.
I want to show that $f(x)$ is irreducible over $R$ if and only if $f(x + r)$ irreducible over $R$.

The map $p(x)\mapsto p(x+r),\,r\in R$ is an automorphism of the ring $R[x]$, and then:

$f(x)$reducible $\Longleftrightarrow f(x)=g(x)q(x),\,g(x),\,q(x)\in R[x]\Longleftrightarrow f(x+r)=g(x+r)q(x+r)$ via the above automorphism $\Longleftrightarrow f(x+r)$ reducible .

Tonio