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Let the restricted direct product $$V_N = (0, \infty) \times \prod_{p \mid N} \{x \in \mathbb Z_p \ : \ x \equiv 1 \mod N\} \times \prod_{p \nmid N} \mathbb Q_p^\times$$

Let $I_N$ be the group al fractional ideals of $\mathbb Q$ prime to $N$. Let $P_N$ be the subgroup of principal fractional ideals $\alpha \mathbb Z$ for $\alpha \in \mathbb Q^\times \cap V_N$. I would like to understand why (and how to make inttuitive sense of) $$I_N /P_N \simeq (\mathbb Z/N \mathbb Z)^\times.$$

I see that there is a natural map $(\mathbb Z/N \mathbb Z)^\times \to I_N$, but why the quotient?

Wolker
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1 Answers1

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$$I_N = \{ \frac{a}b \Bbb{Z}, \gcd(N,ab)=1\}, \qquad P_N = \{ \frac{a}b \Bbb{Z}, \gcd(N,ab)=1, a\equiv b\bmod N\}$$

$$\phi:I_N\to \Bbb{Z}/N\Bbb{Z}^\times/\pm 1, \qquad \phi(\frac{a}b\Bbb{Z})=\pm \frac{a}b $$ is surjective and its kernel is $P_N$, therefore $$I_N/P_N\cong \Bbb{Z}/N\Bbb{Z}^\times/\color{red}{\pm 1}$$

reuns
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