I am currently trying to understand Proposition 2. (page 257 of the JSTOR version) in the above cited paper of Ax. It says the following:
Let $F\supseteq C \supseteq \mathbb{Q}$ be a tower of fields. Then the canonical map $$ C\otimes_{\mathbb{Z}}dF/F \longrightarrow \Omega_{F/C}/dF $$ is injective.
In the proof, it is said that one can assume $F$ has finite transcendence degree over $C$ and the general case can be reduced to the case of transcendence degree $1$. Furthermore, $C$ may be assumed to be relatively algebraically closed in $F$ so that $F=C(T)$ for some $T$ transcendental over $C$.
Now, the part I have trouble figuring out is the following. In the middle of the proof, it is stated:
Thus for each $C$-plane $p$ of $F$ there is a well defined valuation $$ \mathrm{ord}_p: F\longrightarrow \mathbb{Z} \cup \{\infty\} $$ and a $C$-linear map $$ \mathrm{res}_p: \Omega_{F/C} \longrightarrow C. $$
My questions are:
- What is a $C$-plane in this context?
- How does one define valuations associated to such $C$-planes?
- What is the map $\mathrm{res}_p$?