$p(x)$ is irreducible $\iff$ $p(x+c)$ is irreducible.

Question

Let $p(x)$ be a polynomial in $F[x]$, where $F$ is a field. Prove that $p(x)$ is irreducible $\iff$ $p(x+c)$ is irreducible for all $c \in F$.

This question has asked here before, like that:

Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible

However, I didn't understand the argument to prove that $p(x)$ irreducible implies $p(x+c)$ irreducible for all $c \in F$. Also, the otherside you just need to take $c=0$.

So, can you help me how to prove that?

Answer 1

In Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible they have proven that $p(x)$ reducible implies $p(x+c)$ reducible for all $c\in F$.

To prove the same about irreducibility, assume $p(x)$ irreducible. If $q(x)=p(x+c)$ was reducible for some $c\in F$, then $-c\in F$ and so $q(x+(-c))=p(x+(-c)+c)=p(x)$ would be reducible as per their proof - a contradiction.