This question comes from Ramakrishnan's "Fourier Analysis on Number Fields" and its proof of Kronecker-Weber. Namely, I do not understand the final lines of the proof found on p. 236.
Let us denote by $C_\mathbb Q$ the idele class group of $\mathbb Q$, which is $\mathbb I_\mathbb Q/\mathbb Q^*$. We know the open subgroups of $C_\mathbb Q$ are in inclusion-reversing bijective correspondence with finite abelian extensions of $\mathbb Q$ under the Artin map from Artin Reciprocity. The book claims that if you take some open subgroup of $\widehat{\mathbb Z}^\times$, then by examining a neighborhood base of the identity and using the Chinese Remainder Theorem, this open subgroup must contain $U_m$, the open subgroup of $C_\mathbb Q$ associated to the $m$-th cyclotomic extension, for some $m$.
I do not understand what is going on here. Why is this true?