 Cyclotomic Zp-extension

For odd $p$, remove the finite torsion subgroup ${\mu }_{p-1}$ from the Galois group of $\mathbb{Q}({\mu }_{p^{\infty }})/\mathbb{Q}$ by taking its fixed field. For $p=2$, take the fixed field of $\{\pm 1\}$ instead. The remaining Galois group is ${\mathbb{Z}}_p$, using the p-adic logarithm on $1+p{\mathbb{Z}}_p$ or $1+4{\mathbb{Z}}_2$. For a number field $K$, the cyclotomic Zp-extension is the compositum with this rational tower.