Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 24 / 3 / i / Solution

The geometric sum givesSince ${\Phi }_p(1)=p$, its factorization at the nonidentity $p$th roots of unity yieldsEach factor is a unit, and Wilson theorem gives $w\equiv (p-1)!\equiv -1(\mathrm{mod}{\pi }_K)$. Set $u=-w^{-1}$. Then $u$ is a principal unit andThe minus sign comes from the product of the nonzero elements of the residue field, not from an arbitrary choice of uniformizer.