Let $\mathbb{K}$ be a field and $\mathfrak{M}_{\mathbb{K}}$ a family of absolute values on $\mathbb{K}$ with real weights. For each $v\in\mathfrak{M}_{\mathbb{K}}$ we denote by $|\cdot|_{v}$ the corresponding absolute value, by $n_{v}\in\mathbb{R}$ the weight, and by $\mathbb{K}_{v}$ the completion of $\mathbb{K}$ with respect to $|\cdot|_{v}$. We say that $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ is an adelic field if

1. (1)

   for each $v\in\mathfrak{M}_{\mathbb{K}}$, the absolute value $|\cdot|_{v}$ is Archimedean or associated to a nontrivial discrete valuation;

2. (2)

   for each $\alpha\in\mathbb{K}^{\times}$, $|\alpha|_{v}=1$ except a for a finite number of $v$.