I know very little about Iwasawa theory. According to what I know, the way to define the cyclotomic extension ($\mathbf{Z}_p$-extension) $K_{cyc}$ of a number field $K$ is to notice that the Galois group of $K(\zeta_{p^{\infty}})/K$ (adding all $p$-power roots of unity) embeds into $\mathbf{Z}_p^{\ast}$ through the cyclotomic character (recall $\mathbf{Z}_p^{\ast}\simeq\mathbf{Z}/(p-1)\times \mathbf{Z}_p$ for $p>2$ and $\{\pm1\}\times \mathbf{Z}_2$ otherwise). Hence the Galois group is a product of $\mathbf{Z}_p$ with a finite subgroup. Then one can take the fixing field of that finite subgroup and this is $K_{cyc}$.
Question: (1) Can we construct $K_{cyc}$ directly by telling what are the elements added to $K$?
(2) Take $K=\mathbf{Q}$ for example, inside $\mathbf{Q}_{cyc}$ there should be a unique $\mathbf{Q}_1/\mathbf{Q}$ whose Galois group is $\mathbf{Z}/p$ : how to write down this $\mathbf{Q}_1$?