Proposition 7

— Suppose that $K$ is infinite; let $A$ be a commutative $K$-algebra possessing only a finite number of sub-algebras (for example an etale $K$-algebra, $V$, p. 30, Prop. 3) and let $V$ be a vector subspace of $A$ generating $A$. Then there exists $x \in V$ such that $A = K[x]$.

Let $A_1, \ldots, A_n$ be the subalgebras of $A$ distinct from $A$. If $x \notin A_1 \cup \ldots \cup A_n$, then the sub-algebra $K[x]$ cannot equal any of the $A_i$ and so must coincide with $A$. Further, since $V$ generates $A$, it is not contained in any of the subspaces $A_i$. Prop. 7 is therefore a consequence of the following lemma: