(ii) Let P, Q = Z[r]. Prove that P and Q are relatively prime in Q[z]...

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(ii) Let P, Q = Z[r]. Prove that P and Q are relatively prime in Q[z] if and only if the ideal (P,Q) of Zr] generated by P and Q contains a non-zero integer (i.e. Zn(P.Q) {0}). Here (P.Q) is the smallest ideal of Z[r] containing P and Q. (P.Q): (aP+3Qla, 3 € Z[r]). (iii) For which primes p and which integers n 21 is the polynomial a"-p irreducible in Q[z]? Justify your answer. (ii) Let P, Q = Z[r]. Prove that P and Q are relatively prime in Q[z] if and only if the ideal (P,Q) of Zr] generated by P and Q contains a non-zero integer (i.e. Zn(P.Q) {0}). Here (P.Q) is the smallest ideal of Z[r] containing P and Q. (P.Q): (aP+3Qla, 3 € Z[r]). (iii) For which primes p and which integers n 21 is the polynomial a"-p irreducible in Q[z]? Justify your answer.

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Answer Let P Q cz 2 a Suppose P a Contains integer such that Ja B zx Then PQ 2 ... View the full answer