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Rings problems

  1. Mar 11, 2010 #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!
     
  2. jcsd
  3. Mar 11, 2010 #2
    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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