I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for systems of polynomial equations? Say I have a system \begin{cases} f_1(x_1,x_2,\ldots,x_n)=0\\ f_2(x_1,x_2,\ldots,x_n)=0\\ \vdots\\ f_n(x_1,x_2,\ldots,x_n)=0\\ \end{cases} where $f_i(x_1,x_2,\ldots,x_n)=0$ is a polynomial of some degree in the variables $x_1,\ldots,x_n$. (Just to clarify my notation, I am not restricting to homogeneous systems, as $f_i$ may contain also the constant term as well as mixed terms such as $x_1^2x_2^3x_8$ and so on.) My question is, what can I say about the solvability of this system in radicals?

I would be tempted to say that if the total degree of the system is $\geq5$ then the system is not solvable in radicals, because a system of degree $\alpha$ corresponds be a single equation in one variable of degree $\alpha$, but I am not sure if it is right. How would you comment this issue?

alwayssolvable by radicals" to at least fit with the $n=1$ case, or better define "general". $\endgroup$ – YCor Jul 18 at 14:20