p(x)=0,the solutions or root of P(x)
Given a polynomial $p(x)\in \mathbb{Q}[x]$, it is known that its roots can be expressing obtained in terms of the coefficients of the polynomial by Automorphic functionformulas involving the usual algebraic operations (additive,minus,multiplicative,quotient )。May $f(x)$ addition, subtraction, multiplication, division), application of equation radicals (square roots, cube roots, etc), and application of automorphic functions.
What is known when we go up one level, to a polynomial $p_0(x)+p_1(x)f+p_2(x)f^2 +...+p_n(x)f^n=0$ over rational field or p(y)\in \mathbb{Q}(x)[y]$? Meaning, what operations on the extension Q[x] be expressed by Automorphic function too?$p_i(x)f^i$ are all rational Polymonials coefficients (lying in $\mathbb{Q}(x)$) do we need in order to express the roots (lying in $\overline{\mathbb{Q}(x)}$)?
