At the end of Galois Theory we have the theorem:
“Over a field of characteristic zero, a polynomial is solvable by radicals if and only if its Galois group is solvable”
I don’t understand how this connects to being able to find a general formula for the roots of a polynomial of degree $n$ in terms of its coefficients.
I see that for some polynomials of degree $5$, its Galois group is not solvable and so the polynomial cannot be solvable by radicals i.e. its roots are not radical expressions and therefore there must be no general formula for the radical roots of a Quintic equation.
However I don’t understand the converse. If a Galois group of a polynomial $f\in K[x]\setminus\{K\}$, is solvable then the polynomial is solvable by radicals i.e. its roots must be radical expressions of elements in the coefficient field $K[x]$. Good. However the polynomial being solvable by radicals doesn’t imply its roots are radical expressions of precisely the coefficients of $f$.
How then do we know Galois group being solvable means there’s a general radical formula for the roots in terms of the polynomials coefficients?