Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if...

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Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}.

(iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p irreducible in Q[x]? Justify your answer.

(ii) Let P, Q E Z[]. Prove that P and Q are relatively prime in Q[2] if and only if the ideal (PQ) of Z[2] generated by P and

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