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Rings problems

  • #1
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!
 

Answers and Replies

  • #2
236
0
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 [tex]x^m-1[/tex] divides [tex]x^n-1[/tex] if [tex]m\mid n[/tex] (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?
 

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