Editing Factorization (section)

==Ideals==
{{Main|Dedekind domain}}
In [[algebraic number theory]], the study of [[Diophantine equation]]s led mathematicians, during 19th century, to introduce generalizations of the [[integer]]s called [[algebraic integer]]s. The first [[ring of algebraic integers]] that have been considered were [[Gaussian integer]]s and [[Eisenstein integer]]s, which share with usual integers the property of being [[principal ideal domain]]s, and have thus the [[unique factorization domain|unique factorization property]].

Unfortunately, it soon appeared that most rings of algebraic integers are not principal and do not have unique factorization. The simplest example is <math>\mathbb Z[\sqrt{-5}],</math> in which
:<math>9=3\cdot 3 = (2+\sqrt{-5})(2-\sqrt{-5}),</math>
and all these factors are [[irreducible element|irreducible]].

This lack of unique factorization is a major difficulty for solving Diophantine equations. For example, many wrong proofs of [[Fermat's Last Theorem]] (probably including [[Pierre de Fermat|Fermat's]] ''"truly marvelous proof of this, which this margin is too narrow to contain"'') were based on the implicit supposition of unique factorization.

This difficulty was resolved by [[Richard Dedekind|Dedekind]], who proved that the rings of algebraic integers have unique factorization of [[ideal (ring theory)|ideals]]: in these rings, every ideal is a product of [[prime ideal]]s, and this factorization is unique up the order of the factors. The [[integral domain]]s that have this unique factorization property are now called [[Dedekind domain]]s. They have many nice properties that make them fundamental in algebraic number theory.