Exercise 111.21.1. Let $k$ be your favorite algebraically closed field. Below $k[x]$ and $k[x, y]$ denote the polynomial rings.

For every integer $n \geq 0$ find a finite type extension $k[x] \subset A$ of domains such that the spectrum of $A/xA$ has exactly $n$ irreducible components.

Make an example of a finite type extension $k[x] \subset A$ of domains such that the spectrum of $A/(x - \alpha )A$ is nonempty and reducible for every $\alpha \in k$.

Make an example of a finite type extension $k[x, y] \subset A$ of domains such that the spectrum of $A/(x - \alpha , y - \beta )A$ is irreducible

^{1}for all $(\alpha , \beta ) \in k^2 \setminus \{ (0, 0)\} $ and the spectrum of $A/(x, y)A$ is nonempty and reducible.

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