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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."
Further references for the history of ideal numbers.
I don't have the time to add material from these references, but someone else might. These references are quite easy to obtain. At least they should be added at the end of the article in further support of the history part of the article.
Boyer, Merzbach, "A history of mathematics", third edition, John Wiley and Sons, 1968, 1989, 1991, 2011, ISBN 978-0-470-52548-7, pages 521–522, 594.
Florian Cajori, "A history of mathematics", fifth edition, AMS Chelsea Publishing, 1893, 1991, 1999, ISBN 978-0-8218-2102-2, pages 442–445.
Eric Temple Bell, "Men of mathematics", Simon & Schuster, 1937, 1965, 1986, ISBN 978-0-671-62818-5, pages 473–474, 513–514.
Eric Temple Bell, "The development of mathematics", second edition, McGraw-Hill, 1940, 1945, 1967, 1972, 1992, ISBN 978-0-486-27239-9, pages 218, 223–224. — Preceding unsigned comment added by Alan U. Kennington (talk • contribs) 19:00, 13 June 2015 (UTC)