(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In this question we have to make use of the chinese remainder theorem and its applications:

1. Let F be a field and let p1(x), p2(x) two irreducible poynomials such as gcd(p1,p2)=1 over F.

Prove that: F[x]/[p1(x)p2(x)] Isomorphic to F1 x F2 where F1=F[x]/(p1(x)) and

F2=F[x]/(p2(x))...

2. Let F be Z3 (Z/3Z). Find a polynomial p(x)=p1(x)p2(x) such as F[x]/p(x) iso. to F1xF2 where F1,F2 are from order 9, 27 .

2. Relevant equations

3. The attempt at a solution

About 1- in order to prove this using the chinese remainder theo. we need to prove that

p1(x), p2(x) are two relatively prime (coprime) ideals i.e p1(x)+p2(x)=F[x]... But why this is true?

About 2- If I take p1(x)=x^2+1, p2(x)=x^3+2x+1...Both p1 and p2 are irreducible and F1,F2 are from order 9 and 27...gcd(p1,p2)=1 hence we know from 1 that F[x]/[p1(x)p2(x)] Isomorphic to F1 x F2 as needed...

But how can I PROVE 1?

Thanks a lot!

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# Homework Help: Abstract- Polynomial's Ring F[x]

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