# How to compute the $p$-adic period of $\mathbb G_m$?

I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $$\mathbb G_m$$:

I know the example for the infinite place: let $$w=\frac{dz}{z}$$ be the specific holomorphic differential on $$\mathbb G_m(\mathbb C)=\mathbb C^{\times}$$, $$S^1$$ be the unit circle in $$\mathbb C^{\times}$$ which is a generator for $$H_1(\mathbb C^{\times}, \mathbb Z)$$, then the period is the intergal:

$$\int_{S^1} w= \int_{S^1} \frac{dz}{z}=2 \pi i.$$

which is clear from standard complex analysis. My question is how to compute the $$p$$-adic period as the comparison theorem in $$p$$-adic Hodge theory he used:

I think this is a standard exercise in $$p$$-adic Hodge theory, but I couldn't find some references about the computation $$|t_p|_p=p^{-\frac{1}{p-1}}$$ (here the p-adic valuation is normalized such that $$|p|_p=1/p$$). By the way, the original paper is Périodes des variétés abéliennes à multiplication complexe, Ann. of Maths 138 (1993), 625–683, which can be found on Colmez's website.

Thank you for any help.