The determination of the structure of the multiplicative group $({\mathfrak o}/{\mathfrak p}^n)^\times$, which could not be done with Dedekind's theory of ideals, was one of the first successes of Hensel's $\mathfrak p$-adic numbers. You can read all about it in Hasse's Number Theory, Chapter 15, completed by Nakagoshi (Norikata), The structure of the multiplicative group of residue classes modulo ${\mathfrak p}^{N+1}$. Nagoya Math. J. 73 (1979), 41–60, available at http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.nmj/1118785731.

The determination of the structure of the multiplicative group $({\mathfrak o}/{\mathfrak p}^n)^\times$, which could not be done with Dedekind's theory of ideals, was one of the first successes of Hensel's $\mathfrak p$-adic numbers. You can read all about it in Hasse's Number Theory, Chapter 15, completed by Nakagoshi (Norikata), The structure of the multiplicative group of residue classes modulo ${\mathfrak p}^{N+1}$. Nagoya Math. J. 73 (1979), 41–60, available at Link.