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 in the number field there is a positive integer such that is principal. If , then by adjoining any th root of to the number field we go to an extension where unique factorization of ideals shows that , so the ideal number that generates is an th root of an actual number in the domain. Something along those lines might perhaps replace the mention of the Hilbert Class Field.
Second, Harold Edwards and others argue, I think persuasively, that Kummer was mainly investigating higher reciprocity laws rather than Fermat's last theorem when he came up with ideal numbers. Rather than categorically state either one, perhaps something along the lines of "while investigating higher reciprocity laws and also trying to solve Fermat's last theorem."