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

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In summary, the conversation discusses two problems: 1) showing that a ring containing the integers and rational numbers is a principal ideal domain, and 2) proving that a specific polynomial is reducible for every prime number. The second problem involves using a property of roots of unity and considering an extension of degree less than 4.
  • #1
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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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  • #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?
 

1. What is a Principal Ideal Domain (PID)?

A Principal Ideal Domain is a type of ring in abstract algebra where every ideal is generated by a single element. This means that every ideal in a PID can be written as a multiple of a single element within the ring.

2. How can I prove that R is a Principal Ideal Domain?

To prove that R is a PID, you can use the following steps:

  • Show that R is a commutative ring with unity
  • Prove that every ideal in R is a principal ideal
  • Demonstrate that R is an integral domain (i.e. has no zero divisors)

3. What is the significance of X^4+1 being reducible in Z/pZ[X]?

If X^4+1 is reducible in Z/pZ[X], it means that it can be factored into two polynomials of lower degree over the field Z/pZ. This is significant because it shows that the polynomial is not irreducible, meaning it is not a prime element and can be further broken down into smaller factors.

4. How can I prove that X^4+1 is reducible in Z/pZ[X]?

To prove that X^4+1 is reducible in Z/pZ[X], you can use the following steps:

  • Choose a prime number p
  • Show that X^4+1 has a root in Z/pZ
  • Use polynomial long division to factor out the root and show that the resulting polynomial is also reducible in Z/pZ[X]

5. What is the relationship between proving R is a PID and X^4+1 being reducible in Z/pZ[X]?

The relationship is that if R is a PID, then every polynomial in R[X] is either irreducible or can be factored into irreducible polynomials. Therefore, if R is a PID and X^4+1 is reducible in Z/pZ[X], then it can be factored into irreducible polynomials in R[X]. This also means that Z/pZ[X] is a PID, since it is a subring of R[X].

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