Proposition 2 (Hilbert’s Nullstellensatz)

Let k be a field and L an algebraic closure of k.

(i) Every ideal $r \not\subset k[X_1, \ldots, X_n]$ not containing 1 admits at least one zero in $L^n$.

(ii) Let $x = (x_1, \ldots, x_n), y = (y_1, \ldots, y_n)$ be two elements of $L^n$; for the set of polynomials of $k[X_1, \ldots, X_n]$ zero at x to be identical with the set of polynomials of $k[X_1, \ldots, X_n]$ zero at y, it is necessary and sufficient that there exists a k-automorphism s of L such that $y_i = s(x_i)$ for $1 \leq i \leq n$.

(iii) For an ideal a of $k[X_1, \ldots, X_n]$ to be maximal, it is necessary and sufficient that there exist an x in $L^n$ such that a is the set of polynomials of $k[X_1, \ldots, X_n]$ zero at x.

(iv) For a polynomial Q of $k[X_1, \ldots, X_n]$ to be zero on the set of zeros in $L''$ of an ideal c of $k[X_1, \ldots, X_n]$, it is necessary and sufficient that there exist an integer m > 0 such that $Q^m \in c$.