Note that . Don't know if that will help in any way.Let be a polynomial with rational coefficients such that a_n and a_0 are nonzero. Show that p(x) is irreducible over the field of rational numbers if and only if is irreducible over the field of rational numbers.
i am having trouble starting this problem. i know that irreducible means that the polynomial cannot be factored into a product of smaller polynomials. So p(x) = q(x)g(x) + r(x), it will always have a nonzero remainder since it is irreducible. So I am showing the first direction, if p(x) is irreducible then q(x) is irreducible. I tried a contradiction argument assuming q(x) is reducible so there exists g(x) such that q(x) = k(x)g(x). but i can't think of any way to connect that to p(x) and (hopefully) show that p(x) is irreducible as well. Is this an efficient approach? Help is greatly appreciated.