## Irreducibility over Integers mod P

1. The problem statement, all variables and given/known data

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.

2. Relevant equations
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)$$\in$$ 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...

3. 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!
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 Recognitions: Homework Help Science Advisor x^2+1 is reducible mod 5. It's equal to (x+2)(x+3). How can that be when it's roots are imaginary??
 I don't know. I don't know how to prove that they're irreducible, really. I'm just following what somebody said about it being sufficient to show that it has no zeros...

Recognitions:
Homework Help