Editing Algebraic number theory (section)

==History==
===Diophantus===
The beginnings of algebraic number theory can be traced to Diophantine equations,<ref>Stark, pp. 145–146.</ref> named after the 3rd-century [[Alexandria]]n mathematician, [[Diophantus]], who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and the sum of their squares, equal two given numbers ''A'' and ''B'', respectively:

:<math>A = x + y\ </math>
:<math>B = x^2 + y^2.\ </math>

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation <br /> ''x''<sup>2</sup> + ''y''<sup>2</sup> = ''z''<sup>2</sup> are given by the [[Pythagorean triple]]s, originally solved by the Babylonians ({{circa|1800 BC}}).<ref>Aczel, pp. 14–15.</ref> Solutions to linear Diophantine equations, such as 26''x'' + 65''y'' = 13, may be found using the [[Euclidean algorithm]] (c. 5th century BC).<ref>Stark, pp. 44–47.</ref>

Diophantus's major work was the ''[[Arithmetica]]'', of which only a portion has survived.

===Fermat===
[[Fermat's Last Theorem]] was first [[conjectured]] by [[Pierre de Fermat]] in 1637, famously in the margin of a copy of ''Arithmetica'' where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the [[modularity theorem]] in the 20th century.

===Gauss===
One of the founding works of algebraic number theory, the '''''Disquisitiones Arithmeticae''''' ([[Latin]]: ''Arithmetical Investigations'') is a textbook of number theory written in Latin<ref>{{citation |first1=Carl Friedrich |last1=Gauss |first2=William C. |last2=Waterhouse |title=Disquisitiones Arithmeticae |url=https://books.google.com/books?id=DyFLDwAAQBAJ |date=2018 |orig-year=1966 |publisher=Springer |isbn=978-1-4939-7560-0}}</ref> by [[Carl Friedrich Gauss]] in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, [[Euler]], [[Joseph Louis Lagrange|Lagrange]] and [[Adrien-Marie Legendre|Legendre]] and adds important new results of his own. Before the ''Disquisitiones'' was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.

The ''Disquisitiones'' was the starting point for the work of other nineteenth century [[Europe]]an mathematicians including [[Ernst Kummer]], [[Peter Gustav Lejeune Dirichlet]] and [[Richard Dedekind]]. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of [[L-function]]s and [[complex multiplication]], in particular.

===Dirichlet===
In a couple of papers in 1838 and 1839 [[Peter Gustav Lejeune Dirichlet]] proved the first [[class number formula]], for [[quadratic form]]s (later refined by his student [[Leopold Kronecker]]). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general [[number field]]s.<ref name=Elstrodt>{{citation | last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | year = 2007 | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | access-date = 2007-12-25 | archive-date = 2021-05-22 | archive-url = https://web.archive.org/web/20210522140235/https://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | url-status = dead }}</ref> Based on his research of the structure of the [[unit group]] of [[quadratic field]]s, he proved the [[Dirichlet unit theorem]], a fundamental result in algebraic number theory.<ref name=Kanemitsu>{{citation | last = Kanemitsu| first = Shigeru|author2=Chaohua Jia| title=Number theoretic methods: future trends | year=2002| publisher=Springer| isbn= 978-1-4020-1080-4| pages= 271–4}}</ref>

He first used the [[pigeonhole principle]], a basic counting argument, in the proof of a theorem in [[diophantine approximation]], later named after him [[Dirichlet's approximation theorem]]. He published important contributions to Fermat's last theorem, for which he proved the cases ''n''&nbsp;=&nbsp;5 and ''n''&nbsp;=&nbsp;14, and to the [[quartic reciprocity|biquadratic reciprocity law]].<ref name=Elstrodt/> The [[Dirichlet divisor problem]], for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.

===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.

===Hilbert===
[[David Hilbert]] unified the field of algebraic number theory with his 1897 treatise ''[[Zahlbericht]]'' (literally "report on numbers"). He also resolved a significant number-theory [[Waring's problem|problem formulated by Waring]] in 1770. As with [[#The finiteness theorem|the finiteness theorem]], he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.<ref>{{citation |last=Reid |first=Constance |year=1996 |title=Hilbert |publisher=[[Springer Science and Business Media|Springer]] |isbn=0-387-94674-8}}</ref> He then had little more to publish on the subject; but the emergence of [[Hilbert modular form]]s in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on [[class field theory]]. The concepts were highly influential, and his own contribution lives on in the names of the [[Hilbert class field]] and of the [[Hilbert symbol]] of [[local class field theory]]. Results were mostly proved by 1930, after work by [[Teiji Takagi]].<ref>This work established Takagi as Japan's first mathematician of international stature.</ref>

===Artin===
[[Emil Artin]] established the [[Artin reciprocity law]] in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory.<ref>{{citation |author-link=Helmut Hasse |first=Helmut |last=Hasse |chapter=History of Class Field Theory | editor-last=Cassels| editor-first=J. W. S.| editor-link=J. W. S. Cassels| editor2-last=Fröhlich| editor2-first=Albrecht| editor2-link=Albrecht Fröhlich| title=Algebraic number theory| orig-year=1967 |year=2010 |edition=2nd| place=London| publisher=9780950273426| mr=0215665 |pages=266–279}}</ref> The term "[[reciprocity law (mathematics)|reciprocity law]]" refers to a long line of more concrete number theoretic statements which it generalized, from the [[quadratic reciprocity law]] and the reciprocity laws of [[Gotthold Eisenstein|Eisenstein]] and Kummer to Hilbert's product formula for the [[Hilbert symbol|norm symbol]]. Artin's result provided a partial solution to [[Hilbert's ninth problem]].

===Modern theory===
Around 1955, Japanese mathematicians [[Goro Shimura]] and [[Yutaka Taniyama]] observed a possible link between two apparently completely distinct, branches of mathematics, [[elliptic curve]]s and [[modular form]]s. The resulting [[modularity theorem]] (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is [[modular elliptic curve|modular]], meaning that it can be associated with a unique [[modular form]].

It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist [[André Weil]] found evidence supporting it, yet no proof; as a result the "astounding"<ref name="Singh">{{citation |title=[[Fermat's Last Theorem (book)|Fermat's Last Theorem]] |author-link=Simon Singh |first=Simon |last=Singh |year=1997 |publisher=Fourth Estate |isbn=1-85702-521-0}}</ref> conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the [[Langlands program]], a list of important conjectures needing proof or disproof.

From 1993 to 1994, [[Andrew Wiles]] provided a proof of the [[modularity theorem]] for [[semistable elliptic curve]]s, which, together with [[Ribet's theorem]], provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993<ref name=nyt>{{cite news|last=Kolata|first=Gina|title=At Last, Shout of 'Eureka!' In Age-Old Math Mystery|url=https://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html|access-date=21 January 2013|newspaper=The New York Times|date=24 June 1993}}</ref> in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with [[Richard Taylor (mathematician)|Richard Taylor]], and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from [[algebraic geometry]] and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the [[category (mathematics)|category]] of [[scheme (mathematics)|schemes]] and [[Iwasawa theory]], and other 20th-century techniques not available to Fermat.