Is the base change of a smooth curve the same as defining it over a field extension? Let $C$ be a smooth geometrically connected curve defined over a number field $k$. If a closed point $P$ of $C$ is not $k$-rational, i.e., $P \notin C(k)$, then we know that there exists some finite field extension $L/k$ such that $P \in C(L)$. The smallest degree of such a field extension is called the degree of $P$.

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*When we do the base change $C_L := C \times _k L$, is it correct now to say that $P$ is a rational point of $C_L$, i.e., does $P$ have degree $1$ as a closed point of $C_L$?


*If the above is true, then $C_L$ is an $L$-variety. This sort of defines $C$ over the field extension $L/k$, but what is the difference between the closed points of $C_L$ and the closed points of $C$? Are they in bijection with one another?
I apologise if any of these questions follow from definitions or do not make sense, I'm trying to tidy up some concepts here.
 A: (1) We have to be a bit careful in phrasing this, because the points of $C$ are not a subset of the points of $C_L$ in a canonical way, so something like "is $P\in C$ now a closed point of $C_L$" is a bit unclear. Let me try to make sense of it: what we do have is a projection morphism $\pi:C_L\to C$, and then it makes sense to compare a point $P\in C$ with its fiber under $\pi$. The elements of the fiber can then be thought of as 'what $P$ has become in $C_L$', but note that the fiber can have multiple elements.
Now suppose $P$ is rational over a finite field extension $L$ over $k$ (i.e. $L$ is a finite extension of the residue field $\kappa(P)$). Then the elements in the fiber are in bijective correspondence with prime ideals of $\kappa(P)\otimes_k L$ (e.g. by writing $C_L$ as in here). Now we can write $\kappa(P)\cong k[x]/(f)$ as we are in characteristic $0$, and then $f$ splits into linear factors over $L$, so $\kappa(P)\otimes_k L\cong L[x]/(f)$ has exactly $\deg f$ maximal ideals, and all of them have residue field $L$. So in the fiber over $P$ there are exactly $\deg P$ many points, and they are rational over $L$. In particular, they are closed points of $C_L$ seen as a curve over $L$ via the second projection $C_L=X\times_k L\to L$.
(2) Is this clearer now with (1)?
