# Class field theory and the class group

Let $$k$$ be a finite abelian extension of $$\mathbb{Q}$$. Class field theory states that $$k$$ corresponds to some open subgroup of finite index $$U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$$ where $$\mathbb{A}_{\mathbb{Q}}$$ denotes the adeles of $$\mathbb{Q}$$. Specifically the subgroup $$U_k$$ is the image of the idelic norm map. Moreover every open subgroup of finite index arises this way for some unique abelian extension.

The subgroup $$U_k$$ should know everything about the abelian extension $$k$$. So I was wondering, does it know the class group?

Let $$U \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$$ be an open subgroup of finite index with corresponding abelian extension $$k$$. Is there some way to calculate the class group of $$k$$, just knowing $$U$$?

• From $U$ you know some $N$ such that $V_N\subset U$ where $V_N=\prod_{p^k\| N} (1+p^k\mathbb{Z}_p)\prod_{p\nmid N} \mathbb{Z}_p^\times$ then $k = \mathbb{Q}(\zeta_N)^H$ where $H = (U\cap \hat{\mathbb{Z}}^\times)/V_N,Gal(\mathbb{Q}(\zeta_N)/\mathbb{Q}) = \hat{\mathbb{Z}}^\times/V_N$ Feb 10 '19 at 23:27
• The integer $N$ uniquely determines the field $k$ of $N$-th roots of unity, still it does not know its class group in any reasonable way. Feb 11 '19 at 6:04