Residue fields of schemes of finite type over a field $k$.

Question

Let $X$ be a scheme of finite type over a field $k$. Take $x \in X$. Does this imply that 1. $\kappa(x)$ is a field extension of $k$? and 2. It is in fact a finite extension because $X$ is of finite type over $k$? Thank you

Answer 1

1. Yes: $X \to \mathrm{Spec} \ k$ induces for any $x \in X$ maps on residue fields $k \to \kappa(x)$, so that $\kappa(x)$ is a field extension of $k$.

2. This is not true in general; here's a simple counter example: Take $X = \mathbb{A}^1_{k} = \mathrm{Spec} \ k[T] \to \mathrm{Spec} \ k$. Then $X$ certainly is finite type over $k$. Choosing the generic point $\eta = (0)$ of $X$, we have that $\kappa(\eta) = k(T)$ which certainly is not finite over $k$.

As already mentioned in a comment, your statement is true for closed points though because of Zariski's Lemma.