Editing Carl Friedrich Gauss (section)

=== Algebra and number theory ===
==== Fundamental theorem of algebra ====
[[File:DBP 1977 928 Carl Friedrich Gauß.jpg|thumb|upright|German stamp commemorating Gauss's 200th anniversary: the [[complex plane]] or ''Gauss plane'']]
In his doctoral thesis from 1799, Gauss proved the [[fundamental theorem of algebra]] which states that every non-constant single-variable [[polynomial]] with complex coefficients has at least one complex [[root of a function|root]]. Mathematicians including [[Jean le Rond d'Alembert]] had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts led to considerable clarification of the concept of complex numbers.<ref>{{Cite arXiv | last1 = Basu | first1 = Soham | last2 = Velleman | first2 = Daniel J. | date = 21 April 2017 | title = On Gauss's first proof of the fundamental theorem of algebra | class = math.CV | eprint = 1704.06585}}</ref>

==== ''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>

==== Further investigations ====
One of Gauss's first results was the empirically found conjecture of 1792 – the later called [[prime number theorem]] – giving an estimation of the number of prime numbers by using the [[Logarithmic integral function|integral logarithm]].<ref>{{cite book | last1 = Koch | first1 = H. | author-link1 = Herbert Koch | last2 = Pieper | first2 = H. | title = Zahlentheorie | publisher = VEB Deutscher Verlag der Wissenschaften | place = Berlin | date = 1976 | pages = 6, 124}}</ref>{{efn|Gauss told the story later in detail in a letter to [[Johann Franz Encke|Encke]].{{sfn|Bachmann|1922|p=4}}}}

In 1816, [[Heinrich Wilhelm Matthias Olbers|Olbers]] encouraged Gauss to compete for a prize from the French Academy for a proof for [[Fermat's Last Theorem]]; he refused, considering the topic uninteresting. However, after his death a short undated paper was found with proofs of the theorem for the cases ''n'' = 3 and ''n'' = 5.<ref>{{cite journal | last1 = Kleiner | first1 = I. | year = 2000 | title = From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem | journal = [[Elemente der Mathematik]] | volume = 55 | pages = 19–37 | url = http://math.stanford.edu/~lekheng/flt/kleiner.pdf | doi = 10.1007/PL00000079
 | s2cid = 53319514 | url-status = dead | archive-url = https://web.archive.org/web/20110608052614/http://math.stanford.edu/~lekheng/flt/kleiner.pdf | archive-date = 8 June 2011}}</ref> The particular case of ''n'' = 3 was proved much earlier by [[Leonhard Euler]], but Gauss developed a more streamlined proof which made use of [[Eisenstein integers]]; though more general, the proof was simpler than in the real integers case.{{sfn|Bachmann|1922|pp= 60&ndash;61}}

Gauss contributed to solving the [[Kepler conjecture]] in 1831 with the proof that a [[Close-packing of equal spheres|greatest packing density]] of spheres in the three-dimensional space is given when the centres of the spheres form a [[Cubic crystal system|cubic face-centred]] arrangement,<ref>{{Cite journal | last = Hales | first = Thomas C. | author-link = Thomas Callister Hales | title = Historical overview of the Kepler conjecture | doi = 10.1007/s00454-005-1210-2 | mr = 2229657 | year = 2006 | journal = [[Discrete & Computational Geometry]] | issn = 0179-5376 | volume=36 | issue = 1 | pages = 5–20| doi-access = free}}</ref> when he reviewed a book of [[Ludwig August Seeber]] on the theory of reduction of positive ternary quadratic forms.<ref>{{Cite book | last = Seeber | first = Ludwig August | year = 1831 | title = Untersuchungen über die Eigenschaften der positiven ternaeren quadratischen Formen | place = Mannheim  | url = https://books.google.com/books?id=QKJGAAAAcAAJ}}</ref> Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.<ref>{{cite journal | date = July 1831 | title = Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber | url = https://babel.hathitrust.org/cgi/pt?id=mdp.39015064427944&seq=387 | journal = [[Göttingische gelehrte Anzeigen]] | issue = 108 | pages = 1065–1077}}</ref>

In  two papers on [[Quartic reciprocity|biquadratic residue]]s (1828, 1832) Gauss introduced the [[ring theory|ring]] of [[Gaussian integers]] <math>\mathbb{Z}[i]</math>, showed that it is a [[unique factorization domain]],<ref name="Kleiner1998">{{cite journal | url = https://ems.press/journals/em/articles/664 | title = From Numbers to Rings: The Early History of Ring Theory | first1 = Israel | last1 = Kleiner | author-link = Israel Kleiner (mathematician) | journal = [[Elemente der Mathematik]] | volume = 53 | number = 1 | doi = 10.1007/s000170050029 | year = 1998 | pages = 18–35 | zbl = 0908.16001 | doi-access = free}}</ref> and generalized some key arithmetic concepts, such as [[Fermat's little theorem]] and [[Gauss's lemma (number theory)|Gauss's lemma]]. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity<ref name="Kleiner1998" /> – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.<ref>{{cite book | last1 = Lemmermeyer | first1 = Franz | title = Reciprocity Laws: from Euler to Eisenstein | series = Springer Monographs in Mathematics | publisher = Springer | place = Berlin | year = 2000 | page = 15 | isbn = 3-540-66957-4 | doi= 10.1007/978-3-662-12893-0}}</ref>

In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs ([[Gauss sum]]s) can be applied to prove higher reciprocity laws.{{sfn|Bachmann|1922|pp= 52, 57&ndash;59}}