Tower law

Statement

Let $K \subseteq L \subseteq M$ be fields. Then $$[M : K] = [M : L] \cdot [L : K].$$

Remarks

The proof: let $u_1, \dots, u_d$ be a basis of $L$ over $K$, and let $w_1, \dots, w_e$ be a basis of $M$ over $L$. Then the products $u_i w_j$ are a basis of $M$ over $K$. The formula also holds for infinite degrees, as a product of cardinals. So $M/K$ is finite if and only if $M/L$ and $L/K$ are finite. If $[M : K]$ is a prime number, then $M/K$ has no intermediate field other than $K$ and $M$.

Explanations

L1 human

Degrees multiply along a tower of fields. For example, $[\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = 2 \cdot 2 = 4$. The law is like Lagrange's theorem for groups, and it is the main tool to compute degrees.