Is the sum of polynomials solvable by radicals if both polynomials are solvable by radicals?

Question

Is there a specific theorem? I saw it as an extension of a polynomial solvable by radicals plus a constant (as this is solvable via radicals with a simple substitution of y).

Answer 1

The answer is no. An explicit example is as follows. Consider the two polynomials $$f(x) = x^5 + 2$$ $$g(x) = x^2 + 1$$

Clearly both are solvable by radicals, however one may check that $f(x) + g(x)$ has galois group $S_5$, hence is not solvable by radicals.

There are also counterexamples when $f$ and $g$ have the same degree. Consider, for example, $x^5 +1$ and $x^5 + x$.

Answer 2

No. $$x^5+a x^4 + b x^3 + c x^2+ dx + e = (x^5+e) + (a x^4+bx^3+cx^2 + dx)$$