So I'm reading Milne's book on elliptic curves, which is freely available here.
On page 38, he defines a regular map: Let $k$ be a perfect field. A regular map $\varphi : C_{g_1} \to C_{g_2}$ of affine plane curves is a pair of regular functions $(f_1, f_2)$ sending $C_{g_1}(K)$ into $C_{g_2}(K)$ for all fields $K$ containing $k$, i.e. such that for all $K \supset k$, $$ P \in C_{g_1}(K) \implies (f_1(P), f_2(P)) \in C_{g_2}(K) $$
The regular functions on affine plane curves has been defined on page 29, which he denotes by $k[x,y]$. I don't know if his notation is standard, but he writes $k[X,Y]$ for polynomials in two variables $X$ and $Y$, and the small letters $x$ and $y$ are like 'coordinate functions', and thus $k[x,y]$ are polynomial functions on the curve.
To elaborate a bit more, from what I understand, we can view $k[x,y]$ in two ways: 1. Can view it as polynomial functions (with coefficients in $k$) on the affine plane which we denoted as $k[X, Y]$, but restricted to the curve $C_{g_1}$, and the other way is 2. View it as the quotient $k[X,Y] / (g_1(X, Y))$. This is why the small letters $x$ and $y$ are used i think (please correct me if I'm wrong).
So back to the definition of regular maps, I assume that $f_1, f_2 \in k[x,y]$. Now, my confusion is on last equation and statement, the part about 'for all fields $K \supset k$....'. Isn't this always going to be true? Because if $f_1 \in k[x,y]$, and if $P = (a,b) \in K \times K$, then $f_1(P)$ will certainly be in $K$ right? And same for $f_2$. So unless I'm missing something here, I don't see why we must specify the part about 'for all fields $K$..'