Dimension of $K\subset L(\alpha)$ where $L$ is a field extension of $K$

Question

Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a proof. I need hints. Thank you.

Answer 1

No, this is not true in general, take $K=\mathbb{Q}$, $L=\mathbb{Q}(i)$ and $\alpha = \sqrt{2}$.

By tower law, we can say $[K:L(\alpha)]=[K:L][L:L(\alpha)]$ or $[K:K(\alpha)][K(\alpha):L(\alpha)]$, which may or may not be useful depending on what you know about $K,L$ and $\alpha$.

Answer 2

It is true only if $$ [K(\alpha):L(\alpha)]=1, $$ or equivalently $$ K(\alpha)=L(\alpha), $$ since $$ [K:L(\alpha)]=[K:K(\alpha)]\,[K(\alpha):L(\alpha)]. $$