# Irreducible polynomial ring

• March 10th 2010, 08:41 AM
Showcase_22
Irreducible polynomial ring
Quote:

If $f \in \mathbb{Z}[x]$, then define $f_1 \in \mathbb{Z}[x]$ by $f_1(x)=f(x+1)$.

Show that $f$ is irreducible iff $f_1$ is irreducible.
I tried doing it this way:

Suppose $f$ is irreducible.

Then $f(x)=(a_0+a_1x^1+ \ldots +a_nx^n).b$ where $b$ is a unit (ie. $b= \pm 1$).

WLOG, take $b=1$.

Therefore $f_1(x)=f(x+1)=a_0+a_1(x+1)+ \ldots + a_n(x+1)^n$

$= \sum_{i=0}^n a_i+ \left( \sum_{i=1}^n a_i \right).x+ \ldots + a_nx^n$

From here i'm not really sure where to go. Can anyone help?
• March 11th 2010, 01:29 PM
Tinyboss
If $f(x)=g(x)h(x)$, then $f_1(x)=f(x+1)=g(x+1)h(x+1)=g_1(x)h_1(x)$.