Theorem 1

Let $E$ be an extension of $K$ and $A$ an algebra over $E$. Then we have $[A : K] = [A : E] \cdot [E : K]$. In particular, if $F$ is an extension of $E$, we have

The theorem is just a special case of II, p. 222, Prop. 25; more precisely, if $(a_i)$, , , is a basis of $A$ over $E$ and $(b_j)$, $\in M$ a basis of $E$ over $K$, then the family $(a_\lambda b_\mu)_{(\lambda,\mu) \in L \times J}$ is a basis of $A$ over $K$.