Theorem. [1207]

    Let $F \subseteq K \subseteq L$ be fields. Then $[L:F]$ is finite if and only if $[L:K]$ and $[K:F]$ are both finite, and $$ [L:F] = [L : K][K : F]. $$

    Proof. sketch [1207A]

      $\implies$: Given basis of $L$ over $K$ and $K$ over $F$, denote by $\alpha_{i}, 1 \leq i \leq n$ and $\beta_{j}, 1 \leq j \leq m$ respectively, then we can prove $\alpha_{i}\beta_{j} \in L, \forall i, j$ are linearly independent over $F$. It follows that $[L : F] \geq mn$. So if $[L : F] < \infty$, it must follow that $m < \infty$ and $n < \infty$.

      $\impliedby$: Assume now $m, n < \infty$. We can prove that $\alpha_{i}\beta_{j} \forall i, j$ actually spans $L$ over $F$ by showing that every element of $L$ can be written in the form $\sum F \alpha_{i} \beta_{j}$. Then $\alpha_{i} \beta_{j}$ is a basis of $L$ over $F$, hence $[L : F] = mn < \infty$.

      When $[L : F] = \infty$, at most one of $[L:K]$ and $[K:F]$ can be finite. For example, $[\mathbb{C} : \mathbb{Q}] = \infty$ and $[\mathbb{R} : \mathbb{Q}] = \infty$ while $[\mathbb{C} : \mathbb{R}] = 2 < \infty$.