Let f(c) € R[2]. Show that if a € C is a root of f(2), then so is its complex conjugate, a. (You make use without proof that complex conjugation is a ring homomorphism.) (b) Suppose f(1) E Q[2] is irreducible of degree 3. Prove that if the splitting field of f(x) (in C) has degree 3 over Q, then all of the roots of f(2) are real. Hint: By part (a), non-real roots come in pairs. (The Intermediate Value Theorem can also be used to see that f(x) has at least one real root.)
