The solution sheet assumes additional knowledge than what is provided, which annoys me; I don't understand this. Here's the problem
$L:K$ is a field extension. If $\alpha,\beta \in L$ is algebraic, show that $\alpha + \beta $ and $\alpha \beta$ are algebraic.
The theorem used
$L:K$ is a finite extension if and only if $L=K(\alpha_1,...,\alpha_r)$ where $r$ is finite and each $\alpha_i$ are algebraic over $K$.
And the solution says,
The lemma implies that $K(\alpha, \beta):K$ is finite. $K \subseteq K(\alpha\beta) \subseteq K(\alpha, \beta)$ and so by the tower law $[K(\alpha \beta):K]$ is a finite extension and thus $\alpha \beta $ is algebraic.
Woah, wait, we were saying the theorem implies finite degree but then it switches arguments to the Tower Law to conclude it's finite? I don't get it there. How does the tower law tell us it's finite? It only gives us a formula to "calculate" the extension, it doesn't make judgements about finiteness, no?
And, I'm confused, is this solution saying $L=K(\alpha, \beta)$? How do we know that?
A lot of things muddled up in my head, please someone help me out of this abstractness.