Editing Algebraic number theory (section)

===Dedekind===
[[Richard Dedekind]]'s study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as ''[[Vorlesungen über Zahlentheorie]]'' ("Lectures on Number Theory") about which it has been written that:
{{blockquote|"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)}}

1879 and 1894 editions of the ''Vorlesungen'' included supplements introducing the notion of an ideal, fundamental to [[ring (algebra)|ring theory]]. (The word "Ring", introduced later by [[David Hilbert|Hilbert]], does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of [[algebraic integer]]s that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of [[Emmy Noether]]. Ideals generalize Ernst Eduard Kummer's [[ideal number]]s, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem.