How to prove that prime $q \neq p$ isn't ramified in $\mathbb{Z}_p$-extension of $\mathbb{Q}$? I want to show that there is only one $\mathbb{Z}_p$-extension of the field of rational numbers - $\mathbb{Q}$. The cyclotomic one.
For that I need to show that if the field $\mathbb{K}$ is $\mathbb{Z}_p$-extension of $\mathbb{Q}$, then extension $\mathbb{K}/\mathbb{Q}$ is unramified in any prime number $q \neq p$.
I have found two different proof of this fact, but I fail to understand the ending of both of them. One is in the book "L. C. Washington - Introduction to Cyclotomic Fields", on page 265 - Proposition 13.2, and the other one is in the article "On $\Gamma$-extensions of algebraic number fields. Bull. Amer. Math. Soc., 65 (1959), 183-226." of Kenkichi Iwasawa, Lemma 7.1.
 A: For an abelian extension $K/\Bbb{Q}$ then (Kronecker Weber) $Gal(K/\Bbb{Q})$ is a quotient of $Gal(\Bbb{Q}(\zeta_{\infty})/\Bbb{Q}) = \hat{\Bbb{Z}}^\times$.

We assume $Gal(K/\Bbb{Q}) \cong \Bbb{Z}_p$ as profinite topological groups 

$Gal(K/\Bbb{Q})$ is torsion free thus the kernel of $Gal(\Bbb{Q}(\zeta_\infty) / \Bbb{Q}) \to Gal(K/\Bbb{Q})$ contains the profinite completion of $Gal(\Bbb{Q}(\zeta_\infty) / \Bbb{Q})_{tors} = \hat{\Bbb{Z}}^\times_{tors} = \prod_q'(\Bbb{Z}_q^\times)_{tors}$ so that $Gal(K/\Bbb{Q})$ is a quotient of $$ \hat{\Bbb{Z}}^\times/\overline{\hat{\Bbb{Z}}^\times_{tors} } =  \prod_q \Bbb{Z}_q^\times/(\Bbb{Z}_q^\times)_{tors}=(1+4\Bbb{Z}_2)\prod_{q\ne 2}(1+q\Bbb{Z}_q)$$
(think to the RHS not as a subgroup but as a quotient of $\hat{\Bbb{Z}}^\times$)
$Gal(K/\Bbb{Q}) \cong \Bbb{Z}_p$ tells us $K = \bigcup_n K_n$ where $Gal(K_n/\Bbb{Q}) \cong \Bbb{Z}/p^n\Bbb{Z}$, 
Since $Gal(K_n/\Bbb{Q})$ is of order $p^n$ we obtain that $K_n$ is fixed by $$(\hat{\Bbb{Z}}^\times/\overline{\hat{\Bbb{Z}}^\times_{tors} } )^{p^n}=(1+4\Bbb{Z}_2)^{p^n}\prod_{q\ne 2}(1+q\Bbb{Z}_q)^{p^n}= (1+p^{n+1}\Bbb{Z}_p) (1+4\Bbb{Z}_2) \prod_{q\ne 2,p}(1+q\Bbb{Z}_q)$$ and hence $K= \bigcup_n K_n$ is fixed by $$\bigcap_n (\hat{\Bbb{Z}}^\times/\overline{\hat{\Bbb{Z}}^\times_{tors} } )^{p^n}=(1+4\Bbb{Z}_2) \prod_{q\ne 2,p}(1+q\Bbb{Z}_q)=Gal(\Bbb{Q}(\zeta_\infty)/\Bbb{Q}(\zeta_{p^\infty})^{\langle \omega_{p-1}\rangle})$$ Whence $ K $ is a subfield of $ \Bbb{Q}(\zeta_{p^\infty})^{\langle \omega_{p-1}\rangle}$ and since the later is a $\Bbb{Z}_p$-extension, its subfields are finite extensions of $\Bbb{Q}$ and hence $$K=\Bbb{Q}(\zeta_{p^\infty})^{\langle \omega_{p-1}\rangle}$$
