 # Prove that for any two distinct points of an irreducible curve there exists a rational function that Savanah Boone 2022-07-01 Answered
Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other.
I think I can construct such a function, for example, $u\left(x,y\right)=\left(x-a{\right)}^{2}+\left(y-b{\right)}^{2}$ for given two points $\left(a,b\right)$ and $\left(c,d\right)$. However, this doesn't work for general algebraically closed field, for example, the case of $\left(c,d\right)=\left(a+i,b+1\right)$. Hence now I have no clue. Could you give me a hint for this problem?
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Perhaps this is a bit late, but here's what I first thought:
Suppose $a\ne c,b\ne d$, and characteristic is not 2. Then let $u\left(x,y\right):=\frac{x-a}{2\left(c-a\right)}+\frac{y-b}{2\left(d-b\right)}$. Then clearly $u\left(a,b\right)=0$ and $u\left(c,d\right)=\frac{1}{2}+\frac{1}{2}=1$

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