Cyclotomic Milnor square ($\mathbb{Z}[C_p]=\mathbb{Z}{\times }_{{\mathbb{F}}_p}\mathbb{Z}[{\zeta }_p]$)

Evaluation of the cyclic generator at $1$ and at a primitive $p$th root gives the displayed pullback. The maps to the common field reduce integers modulo $p$ and set ${\zeta }_p=1$. If a representative polynomial for the second component has value $c$ at one, adding an integer multiple of ${\Phi }_p$ adjusts that value by a multiple of $p$ without changing the second component. This proves surjectivity onto compatible pairs.