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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.

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The "tower law" usually states that if $L/K/F$ is a tower of field extensions, then $[L:F] = [L:K]\cdot [K:F]$, meaning that if one side is finite then so is the other, and then they are equal. A more sophisticated version would be that they are always equal as cardinals (but that implies knowledge of cardinals).

So in your case, since $K(\alpha,\beta)/K$ is finite, the intermediate extension $K(\alpha\beta)/K$ is also finite (and then $\alpha\beta$ is algebraic over $K$).

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