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

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SUMMARY

The discussion focuses on proving that the ring R, which is contained within the rational numbers Q and includes the integers Z, is a principal ideal domain (PID). The proof leverages the fact that Z is a PID, ensuring that every ideal in Z that is also in R is principal. Additionally, the discussion addresses the reducibility of the polynomial X^4 + 1 in the ring Z/pZ[X] for any prime p, suggesting that if X^m - 1 divides X^n - 1 for m dividing n, then X^4 + 1 can be shown to be reducible by examining its roots in an appropriate field extension.

PREREQUISITES
  • Understanding of principal ideal domains (PIDs)
  • Familiarity with polynomial rings, specifically Z/pZ[X]
  • Knowledge of roots of unity and their properties
  • Basic concepts of field extensions in abstract algebra
NEXT STEPS
  • Study the properties of principal ideal domains and their implications in ring theory
  • Learn about polynomial factorization in finite fields, particularly Z/pZ
  • Explore the concept of roots of unity and their role in polynomial divisibility
  • Investigate field extensions and their applications in proving polynomial reducibility
USEFUL FOR

This discussion is beneficial for students and researchers in abstract algebra, particularly those studying ring theory, polynomial factorization, and field extensions. It is especially relevant for mathematicians focusing on algebraic structures and their properties.

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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!
 
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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?
 

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