Tower Law (Degree Multiplicativity)

Theorem (Tower Law).
Let $K \subseteq L \subseteq M$ be a tower of fields. If $[L:K]$ and $[M:L]$ are finite, then $[M:K]$ is finite and

Equivalently, the degree of an extension is multiplicative in a tower of fields .

Examples.

1. $\mathbb{Q} \subset \mathbb{Q}(\sqrt2) \subset \mathbb{Q}(\sqrt2,\sqrt3)$:
   $[\,\mathbb{Q}(\sqrt2):\mathbb{Q}\,]=2$, $[\,\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}(\sqrt2)\,]=2$, so $[\,\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}\,]=4$.
2. Finite fields: $\mathbb{F}_p \subset \mathbb{F}_{p^2} \subset \mathbb{F}_{p^6}$:
   $[\,\mathbb{F}_{p^2}:\mathbb{F}_p\,]=2$, $[\,\mathbb{F}_{p^6}:\mathbb{F}_{p^2}\,]=3$, hence $[\,\mathbb{F}_{p^6}:\mathbb{F}_p\,]=6$.
3. If $K \subseteq L$ and $M=L$, then $[M:K]=[L:K]\cdot 1$.

Related. field extension , tower law .