I am reading David Zywina's "Elliptic curves with maximal Galois action". For a number field $K$, he defines $\mathbb{Q}^{\text{cyc}} \subset \overline{K}$ to be "the" cyclotomic extension of $\mathbb{Q}$.
- Is $\mathbb{Q}^{\text{cyc}}$ the minimal cyclotomic extension corresponding to the conductor of $K$ obtained from Kronecker-Weber Theorem? What if $K$ isn't abelian?
Further, he defines $\chi_K$ to be the cyclotomic character $\chi_K:G_K \rightarrow \hat{\mathbb{Z}}^{\times}$ where $G_K$ is the absolute Galois group of $K$. He then claims in section $1.1$ that the assumption $K \cap \mathbb{Q}^{\text{cyc}}=\mathbb{Q}$ is equivalent to $\chi_K(G_K)=\hat{\mathbb{Z}}^{\times}$.
- Why is $K \cap \mathbb{Q}^{\text{cyc}}=\mathbb{Q}$ is equivalent to $\chi_K(G_K)=\hat{\mathbb{Z}}^{\times}$?
Any leads would be appreciated!
The extension $\mathbb{Q}^{\rm{cyc}}$ is defined to be $$\mathbb{Q}^{\rm{cyc}} = \bigcup_{n=1}^{\infty} \mathbb{Q}(\zeta_n)$$ where $\zeta_n$ is a primitive $n^{\rm{th}}$ root of unity. Here one obviously has to fix an isomorphism $\overline{\mathbb{Q}} \cong \overline{K}$. One then has $\rm{Gal}(\mathbb{Q}^{\rm{cyc}}/\mathbb{Q}) = \varinjlim \, (\mathbb{Z}/n\mathbb{Z})^\times = \hat{\mathbb{Z}}^\times$ (to see this, choose your roots of unity compatibly).
The second claim is then follows. If $\chi_K(G_K) = H$ is a proper (necessarily open) subgroup of $\hat{\mathbb{Z}}^\times$ then $$K \cap \mathbb{Q}^{\rm{cyc}} = \overline{K}^{\chi_K^{-1}(H)} = (\mathbb{Q}^{\rm{cyc}})^H \neq \mathbb{Q}.$$