Let $k$ be an algebraically closed field, and let $$X = \operatorname{Spec} k[x,y,t]/(ty-x^2)$$ $$Y = \operatorname{Spec} k[t]$$
Hartshorne comments that both schemes $X$ and $Y$ are of finite type over $k$.
This means that the natural morphisms $X \to k$ and $Y \to k$ are of finite type.
I know that if $k$ is algebraically closed then the closed points of $A_k^1$ is in one-to-one correspondence with elements of $k$.
Is $k$ a scheme? How do I show that $X \to k$ a morphism of finite type?