Let be an irreducible polynomial. Form the ideal . Now suppose there is an ideal such that . Since is a PID it means every ideal is principle, thus, . Now since it means , thus, for some . But since is irreducible it means either is constant in that case or is constant in that case . Thus, is a maximal ideal.2. Prove that p in K[X] is irreducible if and only if <p(x)> is a maximal ideal in K[X]
Can you try the converse?