Binary multiplication in Galois Field GF(2^8)

Question

I am working on a project (high school), and I need to explain the process of AES MixColumns for one of the parts.

I am trying to show an example of the matrix multiplication in MixColumns that uses GF(2^8), but I think I did it wrong, and I am not sure where I went wrong. I was wondering if someone could take a look at where my mistake was and explain it to me. Here is what I have written:

I think the mistake may be somewhere in the multiplication of the polynomials (the last step in the picture), but I might be mistaken and it happened earlier.

Note: there's this YouTube video that explains MixColumns in GF(2^3) (since the converted hex to binary is 8 digits), but on all the documentation for AES, they use Rijndaels finite field.

Thanks!

Answer 1

In last step of your calculation:

I think the $x^4$ term is erroneous
My calculation:
$(x^4+x^3+x+1)+x^7+x^6+x^5+x^4+x^2+x$
$=x^7 + x^6 + x^5 + 2 x^4 + x^3 + x^2 +2 x+ 1$
$=x^7 + x^6 + x^5 + x^3 + x^2 + 1$