What is a constant field?

Question

I am looking at the following:

Could you explain to me what a constant field is?

$$$$

P.S. I found this in the paper of T. Honda, "Algebraic differential equation" (pages 170-176).

Answer 1

The word "constant" here is being used in the sense of those elements of $k((x))$ whose differential is zero. Note that in addition to elements $\alpha \in k$, we have power series whose terms have exponents of $x$ that are multiples of $p$ (since this is a finite characteristic of the underlying base field $k$), and by the chain rule, the quotients of two such (nonzero) power series.

The Remark at the bottom of your image has to do with the fact that a reciprocal of a power series $\sum \alpha_i x^i$ in $k[[x]]$ exists in $k[[x]]$ if and only if $\alpha_0 \in k$ is nonzero.

Answer 2

The field of constants of a differential field is the subfield of elements a with $∂a=0$, see here.