Exercise 15, § 14

a) Let K be a field and L a subfield of K such that K is a finitely generated L-algebra, in other words $K = L[a_1, a_2, \ldots, a_n]$ for elements in K. Show that the $a_i$ are algebraic over L. (Argue by contradiction: if $a_1, \ldots, a_m$ ($m \geq 1$) form a maximal algebraically free subfamily of $(a_j)_{1 \leq j \leq n}$, observe that in the ring $A = L[a_1, \ldots, a_n]$ the intersection of all non-zero ideals is reduced to zero, and use Ex. 5 of V, p. 147.) b) Deduce from a) that in the polynomial algebra $K[X_1, \ldots, X_n]$ every maximal ideal is of finite codimension over K. Conclude that if A is a commutative finitely generated algebra over K, then every subfield of A containing K is of finite degree over K.