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Multiplicative representation of algebraic numbers for the range of an arbitrary prime divisor. (Die multiplikative Darstellung der algebraischen Zahlen für den Bereich eines beliebigen Primteilers.) (German) JFM 46.0251.01
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.

##### MSC:
 11S15 Ramification and extension theory 11S99 Algebraic number theory: local and $$p$$-adic fields
##### Keywords:
$$p$$-adic numbers; unit groups; roots of unity
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