# Talk:Ideal number

I would like to propose two changes here: it is unnecessary to invoke the Principalization theorem or the Hilbert Class Field. By finiteness of the class number, for every ideal $I$ in the number field there is a positive integer $n$ such that $I^n$ is principal. If $I^n=(a)$, then by adjoining any $n$th root of $a$ to the number field we go to an extension where unique factorization of ideals shows that $I=(a^{1/n})$, so the ideal number that generates $I$ is an $n$th root of an actual number in the domain. Something along those lines might perhaps replace the mention of the Hilbert Class Field.