Editing Carl Friedrich Gauss (section)

==== ''Disquisitiones Arithmeticae'' ====
{{Main|Disquisitiones Arithmeticae}}
In the preface to the ''Disquisitiones'', Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had independently discovered.{{sfn|Bachmann|1922|p=8}} The ''[[Disquisitiones Arithmeticae]]'', written in 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic [[number theory]]. Therein he introduces the [[triple bar]] symbol ({{math|≡}}) for [[Congruence relation|congruence]] and uses it for a clean presentation of [[modular arithmetic]].{{sfn|Bachmann|1922|pp=8–9}} It deals with the [[unique factorization theorem]] and [[primitive root modulo n|primitive roots modulo n]]. In the main sections, Gauss presents the first two proofs of the law of [[quadratic reciprocity]]{{sfn|Bachmann|1922|pp=16–25}} and develops the theories of [[Binary quadratic form|binary]]{{sfn|Bachmann|1922|pp=14–16, 25}} and ternary [[quadratic form]]s.{{sfn|Bachmann|1922|pp=25–28}}

The ''Disquisitiones'' include the [[Gauss composition law]] for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his [[Legendre's three-square theorem|theorem on three squares]], he proves the triangular case of the [[Fermat polygonal number theorem]] for ''n'' = 3.{{sfn|Bachmann|1922|p=29}} From several analytic results on [[ideal class group|class numbers]] that Gauss gives without proof towards the end of the fifth section,{{sfn|Bachmann|1922|pp=22&ndash;23}} it appears that Gauss already knew the [[class number formula]] in 1801.{{sfn|Bachmann|1922|pp=66&ndash;69}}

In the last section, Gauss gives proof for the [[Constructible polygon|constructibility]] of a regular [[heptadecagon]] (17-sided polygon) with [[Compass and straightedge constructions|straightedge and compass]] by reducing this geometrical problem to an algebraic one.<ref name="Denker">{{cite book | last1 = Denker | first1 = Manfred | last2 = Patterson | first2 = Samuel James | author-link2  = Samuel James Patterson | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 53–62 | chapter = Gauß – der geniale Mathematiker | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> He shows that a regular polygon is constructible if the number of its sides is either a [[power of 2]] or the product of a power of 2 and any number of distinct [[Fermat prime]]s. In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in [[finite field]]s, which amounts to counting integral points on an [[elliptic curve]].<ref name="Stuhler">{{cite book | last1 = Stuhler | first1 = Ulrich | author-link  = Ulrich Stuhler | editor-last = Mittler | editor-first = Elmar | title = "Wie der Blitz einschlägt, hat sich das Räthsel gelöst" – Carl Friedrich Gauß in Göttingen | publisher = Niedersächsische Staats- und Universitätsbibliothek | date = 2005 | series = Göttinger Bibliotheksschriften 30 | pages = 62–72 | chapter = Arithmetisch-geometrisches Mittel und elliptische Integrale: Gauß und die komplexe Analysis | isbn = 3-930457-72-5 | language = de | url = http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/Katalog.pdf}}</ref> An unfinished chapter, consisting of work done during 1797–1799, was found among his papers after his death.{{sfn|Dunnington|2004|p=44}}<ref name="Frei">{{cite book |last=Frei |first=Günther |author-link=Günther Frei |chapter-url=https://books.google.com/books?id=IUFTcOsMTysC&pg=159 |title=The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae |date=2007 |publisher=Springer |isbn=978-3-540-20441-1 |editor-last1=Goldstein |editor-first1=Catherine |editor-link1=Catherine Goldstein |place=Berlin, Heidelberg, New York |pages=159–198 |chapter=The Unpublished Section Eight: On the Way for Function Fields over a Finite Field |doi=10.1007/978-3-540-34720-0 |editor-last2=Schappacher |editor-first2=Norbert |editor-link2=Norbert Schappacher |editor-last3=Schwermer |editor-first3=Joachim |editor-link3=Joachim Schwermer}}</ref>