Let $X=\mathrm{Spec}(\mathbb{Q}[x,y]/(y^2-x^3-x-1))$. So $X$ is an elliptic curve over $\mathbb{Q}$ given by function $y^2=x^3+x+1$.
Now I wonder what is meaning of a $\mathbb{C}$ points of $X$? I think there are two explanations.
Firstly, it can refer a of map $\mathrm{Spec}(\mathbb{C})\to X$.
Secondly, it also may refer a solution of $y^2=x^3+x+1$ in $\mathbb{C}$ such as $(-1,i)$.
I think these two explanations are not equivalent. So what is the definition of a $\mathbb{C}$-point of $X$?
More generally, let $k$ be a field, $k^{'}$ is an field extension of $k$, $X$ is a variety over $k$, $X_{k^{'}}$ is the base change of $X$. If we choose second explanations as our definition, then it seems that the $k^{'}$ points of $X$ are rational points of $X_{k^{'}}$. Am I right?
Thank you very much if you can help me.
There is a bit of notation nightmare, the point is to get the few concepts.
For a prime ideal $I\subset k[x_1,\ldots,x_n]$ and $X=V(I)\subset \Bbb{A}^n_k$ and a field $L\supset k$ then $X(L)=\{ a\in L^n,\forall f\in I, f(a)=0\}$. This is a set. The variety is $X$ which is $X(\overline{k})$ endowed with a topological space structure, a Galois action, plus (more important) its ring of regular functions $k[x_1,\ldots,x_n]/I$ (whose ideals define the open/closed sets of $X(\overline{k})$).
If $I$ is not a prime ideal then $X$ is an algebraic set and the same objects come with it.
The based change is to look at the ideal $I_L= I\ L[x_1,\ldots,x_n]$ (with the prime ideal $I=(x^2+1)\subset \Bbb{Q}[x]$ then $I_\Bbb{C}$ is not a prime ideal anymore) from which we have the algebraic set $X_L=V(I_L)\subset \Bbb{A}^n_L$ which gives again some sets $X_L(E)$ for any field $E\supset L$.
When $I$ is prime and $L=\overline{k}$ or when $L/k$ is Galois it is interesting to look at the action of $Gal(L/k)$ on $X(L)$ and $X_L$ and how it permutes the irreducible components of the latter (equivalently the embeddings of $k[x_1,\ldots,x_n]/I$ into $L[x_1,\ldots,x_n]/I_L$). For $L/k$ Galois then $X(k)$ is the subset of $X(L)$ fixed by $Gal(L/k)$.