> (%i12) factor(x^8+a+1,a^8+a+1);
> Maxima encountered a Lisp error:
>
That's because a^8+a+1 isn't irreducible and therefore isn't a minimum
polynomial defining a unique root as required:
factor(a^8+a+1) => (a^2+a+1)*(a^6-a^5+a^3-a^2+1)
I agree Maxima could give a better error message in this case.
If you use one or the other of these irreducible polynomials, you get good
results:
factor(x^8+a+1,a^2+a+1) =>
(x-a)*(x+a)*(x^2-a-1)*(x^4+a)
factor(x^8+a+1,a^6-a^5+a^3-a^2+1) =>
(x-a)*(x+a)*(x^2+a^2)*(x^4+a^4)