Multiplicative representation of algebraic numbers for the range of an arbitrary prime divisor (Q1470310)

Hensel begins with studying the unit group of the field of \(p\)-adic numbers \(\mathbb Q_p\). Every \(p\)-adic number is a power of \(p\) times a unit, and the unit group is isomorphic to \(\mathbb Z\) times a finite group of roots of unity. The residue classes modulo \(p^s\) of the units form a finite group of order \(\phi(p^s)\) whose structure is determined. In the second section, Hensel treats the corresponding problem for \(p\)-adic number fields, i.e., finite extensions \(K_\pi\) of \(\mathbb Q_p\). Hensel gives a basis for \(1\)-units (these are units \(\equiv 1 \bmod \pi\), where \(\pi\) generates the prime ideal of \(K_\pi\)) and he determines the roots of unity in \(K_\pi\). Editorial comment: Review added in 2016.