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.


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 $K$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.