Proving R is a Principal Ideal Domain and X^4+1 is Reducible in Z/pZ[X]

[rainwyz0706]

1.Let R be a ring such that Z ⊂ R ⊂ Q. Show that R is a principal ideal domain.

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!

 

[Tinyboss]

I know how to do #2, but it's tricky and I'm not sure what would be a useful hint! I'll try...

First convince yourself that x^m-1 divides x^n-1 if m\mid n (think about roots of unity). If you could show that your polynomial divides a different polynomial, one with all its roots in an extension of degree less than 4, what would that mean?