a. Prove that x^2+1 is irreducible over the field F of integers mod 11.
b. Prove that x^2+x+4 is irreducible over the field F of integers mod 11.
c. Prove that F[x]/(x^2+1) and F[x]/(x^2+x+4) are isomorphic.
A polynomial p(x) in F[x] is said to be irreducible over F if whenever p(x)=a(x)b(x) with
a(x),b(x)[tex]\in[/tex] F[x], then one of a(x) or b(x) has degree 0 (i.e. constant).
I was also told by somebody it's sufficient to show that there aren't any zeros...
The Attempt at a Solution
a. The zeros of x^2+1 are + and - i. Therefore, it is irreducible over F.
b. The zeros of x^2+x+4 are also imaginary (-.5 + or - 1.93649167 i), and it is therefore irreducible over F.
c. Each field has 121 elements, so they're isomorphic?
Thanks in advance for the help!