In the field extension $F_p(x)/F_p$, where $F_p(x)$ is the field of fractions of polynomials over $F_p$, is it by definition that $x \in F_p(x)$ is not algebraic? In other words, should I claim that $x$ is indeterminate when I make this field extension, otherwise somehow it is no longer the field of fractions. In other words, if $x \in F_p$ perhaps $F_p(x)$ is a smaller field than the field of fractions.
Further, I read in an answer here: Example of infinite field of characteristic $p\neq 0$ that any element in $F_p(x)$ that involves $x$ will not be algebraic over $F_p$, but I don't see how.