Algebraic Number Theory
《Algebraic Number Theory》 · 第 484–486 页 · Chapter VII，Section 7「Theta Series of Algebraic Number Fields」，开篇与 Proposition 7.1 后定义段
In order to form the theta series we will need, let us now extend $K^*$ to a group $\widehat K^*$ whose elements represent all fractional ideals $\mathfrak a\in J$. The elements $\widehat a$ of $\widehat K^*$ used to be called ideal numbers—a name which is somewhat forgotten but will be used in what follows. The diagram (7.1) implies an isomorphism $\widehat K^*/K^*\cong J/P$. Observe that the ideal numbers are not defined in a canonical way.