<s>[INST] Product in GF(16) i need some help with a product in GF(16), where it is seen as an extension of GF(4)={0, 1, x, x+1} (where $x^2 = x + 1$ ) with the irreducible polynom $f(y) = y^2 + y + x$
So the elements in the fields are: $\{0, 1, x, x+1, y, y + 1, y + x, y + x + 1, xy, ... \}$
I'm try to multiply $((x+1)y + x) * (xy + x + 1)$ and i obtain as result $(x + 1)$ 
I'm not sure that it is the correct result, can you help me?
Thank you,
greetings
 [/INST] Use the identities $x^2 = x+1$, $y^2 = y+x$, and $2=0$. After that you just have to compute. Here it goes explicitly for you to be confident : 
\begin{align}
((x+1)y+x) \times (xy + x + 1) & = ((x+1)y+x)xy + ((x+1)y+x)x + (x+1)y+x \\
& = (x^2y^2+xy^2+x^2y)+(x^2y + xy + x^2) + xy + y + x \\
& = x^2 y^2 + xy^2 + x^2 + y + x \\
& = x^2 (y+x) + x(y+x) + x^2 + (y+x) \\
& = (x^2 + x + 1)(y+x) + x^2 \\
& = (2x^2)(y+x) + x^2 = x^2 = x+1.
\end{align}
If there is a way to make this shorter I haven't looked for it.
Hope that helps,
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