Theorem 2

— Let E be a finitely generated extension of K, generated by the elements $a_1, ..., a_m$, which are algebraic over K; then E is an extension of finite degree of K. If the degree of $a_i$ over $K(a_1, a_2, ..., a_{i-1})$ is $n_i$ (for $1 \leq i \leq m$), then the degree of E over K is $n_1 n_2 ... n_m$ and the elements $a_1^{v_1} a_2^{v_2} ... a_m^{v_m}$ ($0 \leq v_i \leq n_i - 1$) form a basis of E over K.

The elements $a_i^{v_i}$ ($0 \leq v_i \leq n_i - 1$) form a basis of $K(a_1, a_2, ..., a_i)$ over $K(a_1, a_2, ..., a_{i-1})$ by Th. 1, b) of $V$, p. 16; the theorem therefore follows by induction on $m$ from Prop. 25 of II, p. 222.