Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be an adelic field. For $\alpha\in\mathbb{K}^{\times}$, the defect of $\alpha$ is

$$\operatorname{def}(\alpha)=\sum_{v\in\mathfrak{M}_{\mathbb{K}}}n_{v}\log|\alpha|_{v}.$$

Since $\operatorname{def}\colon\mathbb{K}^{\times}\to\mathbb{R}$ is a group homomorphism, we have that $\operatorname{def}(\mathbb{K}^{\times})$ is a subgroup of $\mathbb{R}$. If $\operatorname{def}(\mathbb{K}^{\times})=0$, then $\mathbb{K}$ is said to satisfy the product formula. The group of global heights of $\mathbb{K}$ is $\mathbb{R}/\!\operatorname{def}(\mathbb{K}^{\times})$.