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We show that Z is a principal ideal domain, so every ideal in Z which is also in R is principal. But I'm not sure how to use that R is contained in Q.

2. Proof that X^4+1 is reducible in Z/pZ [X] for every prime p.

I have no clue for this one at all.

Could anyone please offer some insights to either of the above problems? Any help is greatly appreciated!