Let $K$ be a imaginary quadratic field and $K$ has class number one. Let $o$ be ring of integers of $K$. Let $I$ be prime ideal of $o$.
My pdf reads ♯$(o/I)^×＝NI-1$・・・① (My pdf is Rubin's 'Tate Shafarevich groups of elliptic curves with complex multiplication').
My question : What is $N$ ? There is no explanation about this in my pdf, and why does such a equation ① holds ? Thank you for your help.
Since $o$ is an integral extension of $\mathbb{Z}$, it follows that $I\cap\mathbb{Z}$ is a nonzero prime ideal $(p)$ of $\mathbb{Z}$.
Then $o/I$ is a finite integral extension of $F_p$, so $o/I$ is a finite field.
By definition, $N(I)=\text{#}(o/I)$, hence, since $(o/I)^{\times}=(o/I){\setminus}\{0\}$, we get $\text{#}(o/I)^{\times}=N(I)-1$.