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?