So I have the polynomials $f(x) = x^3+x^2+x$ and $g(x)=x^2+x+1$
We are told to find the $gcd$ of both these polynomials in $Q[x], Z/3Z[x], Z/5Z[x], Z/11Z[x]$
After applying the Euclidean algorithm, I see that:
$x^3+x^2+x = x(x^2+x+1) + 0$
Thus the gcd is $x^2+x+1$
However, when I attempt to do the same in all the other fields, I end up with the same answer. I am doing this incorrectly or is it actually possible to have the same gcd in all fields?