I’m reading Silverman’s Arithmetic of Ellipctic curves. In II.4 he gives an definition of differential form as the set of $dx$ for $x$ in $\overline{K}(C)$. Taking the complex circle $x^2+y^2=1$ in $\mathbb{P}^2$. By 4.2(a), $xdy$ is a differential form. By definition there exists $t\in\overline{K}(C)$ such that $dt=xdy$.
If we integrate $\sqrt{1-y^2}dy$, it yields an $arcsin$, which suggests that such a function $t$ wouldn’t be algebraic...
What I’m I missing here? What is the function $t$ such that $dt=xdy$?