Let $F=\mathbb{F}_2(x,y)$ be the Artin-Schreier extension of the rational function field $\mathbb{F}_2(x)$ defined by $y^2+y=\frac{x}{x^3+x+1}$. Calculate the genus of this curve and find the rational places of the extensions $F/\mathbb{F}_2(x)$ and $F_2/\mathbb{F}_4(x)$, where $F_n=F.\mathbb{F}_{q^n}$ is the constant field extension.
A textbook containing this problem says that "..the only ramified place of $\mathbb{F}_2(x)$ in the extension $F/\mathbb{F}_2(x)$ is $x^3+x+1$, and so $F$ has genus $2$.", what could be the possible meaning of this statement? Because I couldn't find any relation between genus and ramified places of an algebraic curve anywhere (or, at least, in that textbook). Moreover, is there any relation between number of ramified places and number of rational places of various extensions of any given function field to determine the same for above extensions given in question? Any help will be appreciated.