ScalingStacks

Lemma 2.46. Let $\boldsymbol{r}=(r_{1},\dots,r_{n})$ be a multi-radius such $\{\alpha(\log r_{i})\}_{i\in\{1,\dots,n\}}$ are $\mathbb{Q}$-linearly independent. Then the $k$-affinoid algebra

$$K_{\boldsymbol{r}}:=k\{\boldsymbol{r}^{-1}\boldsymbol{T},\boldsymbol{r}\boldsymbol{T}^{-1}\}=k\{\boldsymbol{r}^{-1}\boldsymbol{T},\boldsymbol{r}\boldsymbol{S}\}/(T_{1}S_{1}-1,\dots,T_{n}S_{n}-1)$$

is a field. ([Ber, Definition 2.1.1])