Theorem 1 (Weak form)
Let \(X\) be an affine algebraic variety in \(k^{n}\), where \(k\) is an algebraically closed field, and let \(I(X)\) be the ideal of \(X\) in the polynomial ring \(k[t_{1}, \cdots, t_{n}]\). If \(I(X) \neq (1)\) then \(X\) is not empty. Every maximal ideal in the ring \(k[t_{1}, \cdots, t_{n}]\) is of the form \((t_{1} - a_{1},\cdots,t_{n}- a_{n})\) where \(a_{i}\in k\).
Theorem 2 (Strong form)
Let \(k\) be an algebraically closed field, let \(A\) denote the polynomial ring \(k[t_{1}, \cdots, t_{n}]\) and let \(\mathsf{a}\) be an ideal in \(A\). Let \(V\) be the variety in \(k^{n}\) defined by the ideal \(\mathsf{a}\), so that \(V\) is the set of all \(x = (x_{1}, \cdots, x_{n}) \in k^{n}\) such that \(f(x) = 0\) for all \(f \in \mathsf{a}\). Let \(I(V)\) be the ideal of \(V\), i.e. the ideal of all polynomials \(g \in A\) such that \(g(x) = 0\) for all \(x \in V\). Then \(I(V) = r(\mathsf{a})\).
Proof
It is clear that \(r(a) \subseteq I(V)\). Conversely, let \(f \notin r(\mathsf{a})\), then there is a prime ideal \(\mathsf{p}\) containing \(\mathsf{a}\) such that \(f\notin \mathsf{p}\). Let \(\bar{f}\) be the image of \(f\) in \(B = A/\mathsf{p}\), let \(C = B_{f} = B[1/\bar{f}]\), and let \(\mathsf{m}\) be a maximal ideal of \(C\). Since \(C\) is a finitely generated, \(k\)-algebra we have \(C/\mathsf{m} \cong k\), by (7.9)[1]. The images \(x_{i}\) in \(C/\mathsf{m}\) of the generators \(t_{i}\) of \(A\) thus define a point \(x = (x_{1} \cdots, x_{n}) \in k^{n}\), and the construction shows that \(x \in V\) and \(f(x) \neq 0\).
Corollary 1
Let \(k\) be a field and \(B\) a finitely generated \(k\)-algebra. If \(B\) is a field then it is a finite algebraic extension of \(k\).
Corollary 2
Let \(k\) be a field and let \(B\) be a finitely generated \(k\)-algebra. Suppose that \(B\) is a field. Then \(B\) is a finite algebraic extension of k.
Corollary 3
Let \(k\) be afield, \(A\) a finitely generated \(k\)-algebra. Let \(\mathsf{m}\) be a maximal ideal of \(A\). Then the field \(A/\mathsf{m}\) is a finite algebraic extension of \(k\). In particular, if \(k\) is algebraically closed then \(A/\mathsf{m} \cong k\).
Reference
[1] Michael Atiyah. Introduction to commutative algebra. CRC Press, 2018.