Lemma 1

Let $A$ be a finitely generated algebra over a field $k$, $(a_i)_{1 \leq i \leq n}$ a system of generators of this algebra and $c$ the ideal of algebraic relations between the $a_i$ with coefficients in $k$ (Algebra, Chapter IV, § 2, no. 1). For every extension field $L$ of $k$, the mapping $f \mapsto (f(a_i))_{1 \leq i \leq n}$ is a bijection of the set of k-homomorphisms from $A$ to $L$ onto the set of zeros of $r$ in $L^n$.

There exists a unique k-algebra homomorphism $h$ of $k[X_1, \ldots, X_n]$ onto $A$ such that $h(X_i) = a_i$, for $1 \leq i \leq n$ and by definition $r$ is the kernel of $h$. The mapping f $\mapsto$ his a bijection of the set of k-homomorphisms from $A$ to $L$ onto the set of k-homomorphisms from $k[X_1, \ldots, X_n]$ to $L$ which are zero on c. For every polynomial $P \in k[X_1, \ldots, X_n]$ and every element $x = (x_1, \ldots, x_i) \in L^n$ we write $h_x(P) = P(x)$; then the mapping $x \mapsto h_x$ is a bijection of $L^n$ onto the set of k-homomorphisms from $k[X_1, \ldots, X_n]$ to $L$ (such a homomorphism being determined by its values at the $X_i$ ($1 \leq i \leq n$) ); to say that $h_x$ is zero on c means that $x$ is a zero of c in $L''$, whence the lemma.

If Proposition 1 is applied to the algebra $A = k[X_1, \ldots, X_n]/r$, where c is an ideal of $k[X_1, \ldots, X_n]$ distinct from the whole ring, we obtain by Lemma 1 the following statement: