Give examples of prime and maximal ideals in \(k[x,y]\), describe the corresponding varieties, and state the Nullstellensatz.
Solution
Let \(k\) be a field.
1. Prime ideal example.
The ideal \[ (x)\subset k[x,y] \] is prime because \[ k[x,y]/(x)\cong k[y] \] is an integral domain. Its affine variety is the \(y\)-axis \[ V(x)=\{(0,b):b\in k\}. \]
2. Maximal ideals over an algebraically closed field.
Assume now that \(k\) is algebraically closed. For every \((a,b)\in k^2\), \[ \mathfrak m_{a,b}=(x-a,y-b) \] is maximal because \[ k[x,y]/\mathfrak m_{a,b}\cong k. \] The weak Nullstellensatz says that these are all maximal ideals of \(k[x,y]\).
3. Strong Nullstellensatz.
For any ideal \(I\subseteq k[x_1,\dots,x_n]\) with \(k\) algebraically closed, \[ I(V(I))=\sqrt I. \] Equivalently, a polynomial vanishes on every common zero of \(I\) exactly when some power of it lies in \(I\).