I'm currently studying Galois Theory using Fraleigh's abstract algebra. when proving the insolvability of the quintic, the book takes five independent transcendental elements over $\mathbb{Q}$ and forms a "general quintic equation" and a splitting field, then shows that its galois group over $\mathbb{Q}(s_1, ..., s_5)$ is isomorphic to $S_5$. The proof was pretty straightforward.
But I don't quite see why this has to do with solving quintics. Why take transcendental numbers? After all, all the zeroes of a quintic must be algebraic by definition. I'm not even sure if you can call it a "general" quintic, if all of its zeroes are transcendental. I'm sure I am missing something (maybe trivial) here, but I can't just figure it out. Could someone please explain, how this kind of proof can be valid?