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This might be super elementary but, after having read abstract concepts of algebraic curves, I have trouble dealing with actual examples. For instance, why is the $\phi=\dfrac{y}{x}$ a rational function on the curve $F=y^2+y+x^2$? I know that any rational function on this curve should be of the form $\{\phi=\dfrac{f}{g}:f,g\in K[x,y]/(F), g\neq 0\}$, but what do I need to actually check to show that this is a rational function on $F$?

Also, if anybody can give more examples of this kind I will be very grateful.

Thanks for your help in advance.

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    $\begingroup$ Let $U$ be a subset of $\overline{K}$, can you define the ring of polynomial functions $U\to \overline{K}$ and the subring of those given by polynomials with coefficients in $K$ ? What happens if $U=V(I)$ with $I$ a prime ideal of $K[X_1,\ldots,X_n]$ ? What is the ring of $K$-rational functions $U\to \overline{K}$ then ? $\endgroup$
    – reuns
    Oct 17, 2019 at 22:11

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Your definition of a rational function is just fine for where you're at and your function exactly fits it. To see this, note that you need to verify that $f=y$ and $g=x$ are elements of $K[x,y]/(F)$ and that $g\neq 0$ (as a function). But this is clear.

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