Discrete valuation on $\mathbb{F}_p[[t]]$

Question

I know $\mathbb{F}_p[[t]]$ is DVR, but I wonder what valuation are on it. Could you tell me discrete valuation on $\mathbb{F}_p[[t]]$? In other word, could you tell me an example of value from $\mathbb{F}_p[[t]]$ to $\mathbb{R}$ whose value group is $\mathbb{Z}$?

Answer 1

It’s quite easy, really, and I believe that there’s only the one discrete valuation:

If $\,0\ne f\in\Bbb F_p[[t]]$, i.e. $f=\sum_{i=0}^\infty a_it^i$, then there’s a unique $n\ge0$ such that $f=t^ng$, where $g$ has nonzero coefficient in degree $0$. Then $v(f)=n$. Alternatively, you can define $n$ as the lowest degree of a nonzero monomial in $f$.

Answer 2

There is an internal ring theoretic definition of a DVR as a local PID (usually we assume it's also not a field). In such a ring, if the unique maximal ideal is $(\pi)$ we refer to $\pi$ as a uniformizer for the DVR, and this gives us an associated discrete valuation. I'll leave the general case and return to this power series ring, but keep in mind that what I'm going to do will generalize.

Actually, I'll work in the slightly greater generality of an arbitrary field $k$, mostly to save myself some writing. Anyways, in the above jargon we have that $t$ is a uniformizer of $k[[t]]$. Indeed, I won't give the proof, but recall that a formal power series is invertible iff its constant coefficient is a unit. Now, given any formal power series $0 \neq f = \sum a_i t^i$ we factor out the largest power of $t$ we can, leaving us with $f = t^e g$ where $g(0) \neq 0$. We then take $v(f) = e$. This is often called the $t$-adic valuation. I'll leave it to you to verify that this is actually a valuation.

This valuation essentially measures the degree of vanishing of the power series $f$ at the point $0$, i.e. the multiplicity of the root at $0$. You can think of it, for instance, as the smallest $i$ such that $a_i \neq 0$. Also, if $k$ has characteristic $0$ you can define a nice formal differentiation operator $D$ which sends $t^n \mapsto n t^{n-1}$. Of course we can define this in positive characteristic but it vanishes often. With this in mind, the notion of order of vanishing aligns itself with calculus, i.e. saying that $v(f) = e \geq 1$ says that $f$ along with its derivatives up to order $e - 1$ vanish at $0$, but $D^e f(0) \neq 0$.

This idea comes up a lot, so it's worth thinking about the general case of a $\pi$-adic valuation. For example, the $p$-adic integers have the $p$-adic valuation. The above interpretation of a valuation as an "order of vanishing" has real meaning here too. Also, in algebraic geometry, we can consider some "nice" curve $X$ (think of a Riemann surface), a point $P \in X$, and a regular function $f$ defined near $P$ (think holomorphic). Then indeed, we can consider a valuation $v_P(f)$ that returns the order of vanishing of $f$ at $P$. When we pass to rational (meromorphic) functions, this detects the order of poles as well.