Hello. I worked through a Galois field problem and I just wanted to see if I have done everything correctly.

The question follows,

Compute in GF(2^8):

(x^4 + x + 1)/(x^7 + x^6 + x^3 + x^2)

where the irreducible polynomial is P(x) = x^8 + x^4 + x^3 + x + 1.

The first thing I did was find the inverse of (x^7 + x^6 + x^3 + x^2) which equals (x^4 + x^3 + x + 1).

Then, I did (x^4 + x + 1)*(x^4 + x^3 + x + 1)

=x^8 + x^7 + x^5 + x^4 + x^4 + x^2 + x + x^4 + x^3 + x + 1

=x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + 1

I declared x^8 = x^7 + x^5 + x^4 mod P(x)

I plugged this into the result I have.

=(x^7 + x^5 + x^4) + x^7 + x^5 + x^4 + x^3 + x^2 + 1

=x^3 + x^2 + 1 (Final Answer)

Can anyone just check my work to see if I have done everything correctly? If I am wrong, I am not looking for the actual answer, but maybe if you can point out the section where I did something wrong would suffice. Thank you very much in advance.