'shifted' polynomial preserves validity of Eisenstein irreducibility criterion of original, in finite fields?

Question

This is not really research level, but I know not where else to ask it.

The Eisenstein criterion for polynomial irreducibility over rationals or integers permits shifting the original (primitive) polynomial by substituting (x + a) in place of the original variable x, for some integer a. If the shifted polynomial is irreducible, then so was the original, since this shifting is an automorphism on the ring of polynomials over the rationals.

Is the same shifting permitted over finite fields, such that the irreducibility of the shifted polynomial p(x) over finite field GF(q) with a < q guarantees the irreducibility of the original polynomial?

Also, a second question, somewhat related, if a polynomial is irreducible over Z, is it irreducible over any finite field?

Answer 1

If $F$ is any field, and $p$ is any polynomial over $F$, and $a$ is any element of $F$, then $p(x)$ is irreducible over $F$ if and only if $p(x+a)$ is irreduible over $F$. This has nothing to do with Eisenstein. The proof will be a good exercise for you.

EDIT: $x^2+1$ is irreducible over the integers but not over the integers modulo $2$.

Answer 2

For the first question, the answer is of course yes, because if $q(x) = p(x+a)$, then $p(x) = q(x-a)$, so if $q(x)=r(x)s(x)$, then $p(x)=r(x-a)s(x-a)$.

For the second question, the answer is no, for example $x^2+1$ is irreducible over $\mathbb{Z}$, but $(x+1)(x+1) = x^2 + 1$ over $\mathbb{Z}_2$.