Worked solution: Abel–Ruffini theorem and the Galois solvability criterion (1824)

The "general" (or generic) quintic treats $a_0,\ldots,a_4$ not as specific numbers but as independent indeterminates over $\mathbb{Q}$; equivalently, one may start from five independent indeterminates $x_1,\ldots,x_5$ meant to be the roots and let the $a_i$ be (up to sign) the elementary symmetric functions of the $x_i$, so $F=\mathbb{Q}(a_0,\ldots,a_4)$ sits inside $\mathbb{Q}(x_1,\ldots,x_5)$. Because every permutation of $x_1,\ldots,x_5$ fixes each elementary symmetric function (hence fixes $F$), and a classical theorem of Lagrange shows these are exactly all the automorphisms fixing $F$, the Galois group of the splitting field $\mathbb{Q}(x_1,\ldots,x_5)$ over $F$ is the full symmetric group $S_5$ on the five roots — not some smaller subgroup, as can happen for special numerical quintics.