division of polynomials in rings

• Nov 2nd 2010, 08:02 PM
Silverflow
division of polynomials in rings
Hey,
I don't know how to do the following question:

For which integers $n$ does $x^2+x+1$ divide $x^4+3x^3+x^2+6x+10$ in $\mathbb{Z}/n\mathbb{Z}[x]$?

I'm afraid I'm not entirely sure what to do for this question. I think my confusion for this question begins with the definition of $\mathbb{Z}[x]$. Is $\mathbb{Z}[x]$ the ring of algebraic integers? Should I be looking an integer $x$ that makes the 2 above polynomials equal to 0, then find out what integer multiples make the 2 polynomials equal to 0 as well?

• Nov 2nd 2010, 08:26 PM
tonio
Quote:

Originally Posted by Silverflow
Hey,
I don't know how to do the following question:

For which integers $n$ does $x^2+x+1$ divide $x^4+3x^3+x^2+6x+10$ in $\mathbb{Z}/n\mathbb{Z}[x]$?

I'm afraid I'm not entirely sure what to do for this question. I think my confusion for this question begins with the definition of $\mathbb{Z}[x]$. Is $\mathbb{Z}[x]$ the ring of algebraic integers? Should I be looking an integer $x$ that makes the 2 above polynomials equal to 0, then find out what integer multiples make the 2 polynomials equal to 0 as well?

I think they meant $\left(\mathbb{Z}/n\mathbb{Z}\right)[x]=$ the ring of polynomials over the ring of integers modulo n.