According to the entry ``fractional ideal'', we can state that in a Dedekind domain $R$ , each non-zero integral ideal $\mathfrak{a}$ may be written as a product of finitely many prime ideals $\mathfrak{p}_i$ of $R$ , $$\mathfrak{a} = \mathfrak{p}_1\mathfrak{p}_2...\mathfrak{p}_k.$$ The product decomposition is unique up to the order of the factors. This is stated and proved, with more general assumptions, in the entry ``prime ideal factorisation is unique''.

Corollary. If $\alpha_1$ , $\alpha_2$ , ..., $\alpha_m$ are elements of a Dedekind domain $R$ and $n$ is a positive integer, then one has

(1)

This corollary may be proven by induction on the number $m$ of the generators (not on the exponent $n$ ).