关于Serre《代数曲线与类域》中代数曲线有理微分迹定义的若干技术问题咨询

我最近在研读Serre的《Algebraic Curves and Class Fields》第二章第12节时，遇到了关于有理微分形式（也叫亚纯微分形式）迹定义的几个技术问题，先把相关原文摘录如下（第22页内容）：

Now let $X$ be any curve. We choose a function $\varphi$ on $X$ which is not constant. If $X^{\prime}$ denotes the projective line $\mathbf{P}_1(k)$, we can consider $\varphi$ as a map $X \rightarrow X^{\prime}$ which is evidently surjective; it makes $X$ a "covering" of $X^{\prime}$, possibly ramified. Putting $E=k\left(X^{\prime}\right)$ and $F=k(X)$, the map $\varphi$ defines an embedding of $E$ in $F$; the field $E$ is thus identified with the field $k(\varphi)$ generated by $\varphi$. Since $X$ has dimension $1,\left[F: F^p\right]=p$; if $F^{\prime}$ denotes the largest separable extension of $E$ contained in $F$, there thus exists an integer $n \geq 0$ such that $F^{\prime}=F{p^n}$. The extension $F / E$ is separable if and only if $n=0$, in other words if $\varphi \notin F^p$; we assume this from now on.

If $f$ is an element of $F$, its trace in $F / E$ is well defined; it is an element of $E$ which we will write $\operatorname{Tr}_{F / E}(f)$. The operation of trace can be extended to differentials in the following way:

The injection $E \rightarrow F$ defines a homomorphism from $D_k(E)$ to $D_k(F)$; as $d \varphi$ is an $E$-basis of $D_k(E)$ and $\varphi \notin F^p$, this homomorphism is injective and extends to an isomorphism of $D_k(E) \otimes_E F$ with $D_k(F)$. On the other hand, $\operatorname{Tr}{F / E}: F \rightarrow E$ is $E$-linear; applying this homomorphism to the second term of $D_k(E) \otimes_E F$, we finally deduce an $E$-linear map
$$
\operatorname{Tr}{F / E}: D_k(F) \rightarrow D_k(E) .
$$
We can make this more explicit as follows: if $\omega$ is a differential on $X$, we write $\omega=f d \varphi$ and then
$$
\operatorname{Tr}{F / E}(\omega)=\left(\operatorname{Tr}{F / E}(f)\right) d \varphi.
$$
Thus, to every differential $\omega$ on $X$ we have associated a differential $\operatorname{Tr}(\omega)$ on $X'=\mathbf{P}_1(k)$.

我的疑问如下：

• 在第二句中，“a function $\varphi$ on $X$ which is not constant”具体指什么？是不是就是$F\setminus k$中的元素？
• 如果$\varphi$确实是$F\setminus k$中的元素，那它是如何诱导出映射$X\to X'$的？为什么说这个映射“evidently surjective”（显然是满射）？
• 为什么“$X$ has dimension $1$”能推出$\left[F: F^p\right]=p$？我完全不了解弗罗贝尼乌斯相关的内容，搜索“frobenius degree”也没找到有用的信息。
• 在第三段第一句中，怎么从$\varphi\notin F^p$推出$d\varphi\neq 0$？

另外需要说明：Serre在他的书中始终假设$\overline{k}=k$，并且“簇”（特别是“曲线”）指的是“FAC意义下的簇”——即局部是有限生成既约$k$-代数的极大谱的$k$-局部环空间，且是分离的。

备注：内容来源于stack exchange，提问作者Elías Guisado Villalgordo