On the structure of \((O_k/I)\) (Q5929410)

Dirchlet's unit theorem describes the group structure of the group \({\mathfrak o}_K^\times\) of all global units in a number field \(K\). This paper studies the group structure of the finite Abelian groups \(({\mathfrak o}_K/{\mathfrak a})^\times\) where \({\mathfrak a}\) is a nonzero ideal in \({\mathfrak o}_K\), provided that the prime factors \({\mathfrak p}\) of \({\mathfrak a}\) have absolute ramification indices \(e< p\overset {\text{def}}= \text{char}({\mathfrak o}_k/{\mathfrak p})\). The structure depends on whether \(e+1< p\) always holds or not. In any case, for \({\mathfrak a}\) a power of \({\mathfrak p}\), it is determined by the splitting type of \({\mathfrak p}\). The restriction on the size of \(e\) is due to the behaviour of the \(p\)-power homomorphism on the local unit groups \(U_{\mathfrak p}^{(n)}\).