Suppose $k$ is a field of characteristic 0, and $K$ is a finite extension of $k$. If $X$ is a variety defined over $K$, and by restriction of base field $X$ is also a variety over $k$.
Question 1: What are the differences between the subvarieties of $X$ considered as a variety over $K$ and over $k$?
Question 2: How does the Chow group $CH^*(X)$ change when we restrict the base field from $K$ to $k$?