Theorem (Arnold - 1970): The algebraic function defined by the equation $$\ \ z^n+a_1z^{n-1}+\cdots +a_{n-1}z+a_n\ \ \ ,$$ i.e. the image of $X$ in $$\mathbf C[a_1,\dots,a_n][X]\ /\ (X^n+a_1X^{n-1}+\cdots +a_{n-1}X+a_n)$$ cannot be written as a composition of polynomial functions of any number of variables and algebraic functions of less than $\phi(n)$ variables, where $\phi(n)$ is $n$ minus the number of ones appearing in the binary representation of the number $n$.

The proof is essentially a clever application of the computation of the mod. 2 cohomology ring of the braid group $B_n$ by Fuchs.

(And I seem to remember Vershinin explaining that Arnold asked Fuchs to compute this ring for this very reason).

Theorem (Arnold - 1970): The algebraic function defined by the solutions of the equation $$\ \ z^n+a_1z^{n-1}+\cdots +a_{n-1}z+a_n=0\ \ \ ,$$ cannot be written as a composition of polynomial functions of any number of variables and algebraic functions of less than $\phi(n)$ variables, where $\phi(n)$ is $n$ minus the number of ones appearing in the binary representation of the number $n$.

The proof is essentially a clever application of the computation of the mod. 2 cohomology ring of the braid group $B_n$ by Fuchs.

(And I seem to remember Vershinin explaining that Arnold asked Fuchs to compute this ring for this very reason).