absolute value and valuation on the field of formal power series

Question

I'm currently working on (non-Archimedean) valuations and absolute values. My text states that one example, where I have both a non-Archimedean valuation and a corresponding absolute value is $\mathbb{C}((t))$ , i.e. the field of formal power series with coefficients in $\mathbb{C}$, defined as $$\mathbb{C}((t))= \{ \sum_{k=m}^\infty a_kt^k : m \in \mathbb{Z}, a_k \in \mathbb{C}\}.$$

I tried to find how a valuation and an absolute value on this field are defined, but I came up empty-handed. It would be great if any of you could help me! (If it helps - they mentioned in the same context also $\mathbb{Q}_p$ with the $p$-adic valuation/ absolute value and the completion $\mathbb{C}_p$).

Answer 1

The valuation $v(f(t))$ is given by the number of times $t$ divides the power series $f(t)$. The norm is then defined accordingly, i.e. one may define it by setting for example $|f(t)| := e^{-v(f(t))}$. (More generally, this kind of construction whenever you take a Dedekind domain and consider some nonzero prime ideal of it. In this case the ring is $\mathbb{C}[[t]]$ and the ideal is $(t)$. The case where the domain is $\mathbb{Z}$ and the ideal is $(p)$ gives the usual case of $\mathbb{Q}_p$).