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{{Infobox scientist
|name              = Ludwig Stickelberger
|image             =
|image_size        =
|caption           = 
|birth_date        = May 18, 1850
|birth_place       = [[Buch, Schaffhausen]]; 
|death_date        = April 11, 1936
|death_place       = [[Basel]]
|residence         = 
|citizenship       = 
|nationality       = [[Switzerland|Swiss]] 
|ethnicity         = 
|field             = [[Mathematics]]
|work_institutions = [[University of Freiburg]] 
|alma_mater        = [[University of Heidelberg]] <br> [[Humboldt University of Berlin|University of Berlin]]
|doctoral_advisor  = [[Karl Weierstrass]]
|doctoral_students = 
|known_for         =  [[Stickelberger's theorem|Stickelberger relation]]
|author_abbrev_bot = 
|author_abbrev_zoo = 
|influences        = 
|influenced        = 
|prizes            = 
|religion          = 
|footnotes         = 
|signature         =
}}

'''Ludwig Stickelberger''' (May 18, 1850 &ndash; April 11, 1936) was a [[Swiss]] [[mathematician]] who made important contributions to [[linear algebra]] (theory of [[elementary divisors]]) and [[algebraic number theory]] (Stickelberger relation in the theory of [[cyclotomic field]]s).

==Short biography==
Stickelberger was born in Buch in the [[canton of Schaffhausen]] into a family of a pastor. He graduated from a gymnasium in 1867 and studied next in the [[University of Heidelberg]]. In 1874 he received a doctorate in [[Berlin]] under the direction of [[Karl Weierstrass]] for his work on transformation of [[quadratic form]]s to a diagonal form. In the same year he obtained his [[Habilitation]] from Polytechnicum in [[Zurich]] (now [[ETH Zurich]]). In 1879 he became an extraordinary professor in the [[University of Freiburg|Albert Ludwigs University of Freiburg]]. From 1896 to 1919 he worked there as a full professor, and from 1919 until his return to Basel in 1924 he held the title of a distinguished professor ("ordentlicher Honorarprofessor"). He was married in 1895, but his wife and son both died in 1918. Stickelberger died on April 11, 1936 and was buried next to his wife and son in Freiburg.

==Mathematical contributions==
Stickelberger's obituary lists the total of 14 publications: his thesis (in Latin), 
8 further papers that he authored which appeared during his lifetime, 4 joint papers with [[Georg Frobenius]] and a posthumously published paper written circa 1915. Despite this modest output, he is characterized there as "one of the sharpest among the pupils of Weierstrass" and a "mathematician of a high rank". Stickelberger's thesis and several later papers streamline and complete earlier investigations of various authors, in a direct and elegant way.

===Linear algebra===
Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and [[Gaston Darboux|Darboux]]. Augmented with the contemporaneous work of Frobenius, it set the theory of [[elementary divisors]] upon a rigorous foundation. An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the [[classification of finitely generated abelian groups]] and sketched the relation with the theory of [[module (mathematics)|modules]] that had just been developed by [[Richard Dedekind|Dedekind]].

===Number theory===
Three joint papers with Frobenius deal with the theory of [[elliptic function]]s. Today Stickelberger's name is most closely associated with his 1890 paper that established the [[Stickelberger relation]] for cyclotomic [[Gaussian sum]]s. This generalized earlier work of [[Carl Gustav Jacob Jacobi|Jacobi]] and [[Ernst Kummer|Kummer]] and was later used by [[David Hilbert|Hilbert]] in his formulation of the reciprocity laws in [[algebraic number field]]s. The Stickelberger relation also yields information about the structure of the [[class group]] of a [[cyclotomic field]] as a module over its abelian [[Galois group]] (cf [[Iwasawa theory]]).

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Seine Untersuchungen betreffen unterschiedliche Bereiche der Algebra (beispielsweise mit seinem Freund  [[Ferdinand Georg Frobenius]] über Gruppentheorie), Analysis und Zahlentheorie, in denen er sich als „als einer der scharfsinnigsten Schüler von Weierstraß“ zeigte (Heffter in seinem Nachruf). Bekannt sind vor allem seine Arbeiten zum Kreisteilungskörper (''Ueber die Verallgemeinerung der Kreistheilung''). Dort liefert der „Satz von Stickelberger“ explizit formale Summen <math>S</math> von Elementen der Galoisgruppe <math>G = \{\sigma_a | a \in (\mathbb Z / p\mathbb Z)^*\} = \{\sigma_a | a \in (1, \ldots, (p-1))\}</math> des [[Kreisteilungskörper]]s <math>\mathbb{Q}(\zeta_p)</math>, die auf ein beliebiges [[Ideal (Mathematik)|Ideal]] <math>I</math> im [[Ring (Mathematik)|Ring]] der ganzen Zahlen <math>\mathbb{Z}(\zeta_p)</math> des Kreisteilungskörpers angewandt ein Hauptideal ergeben. <math>S</math> „annihiliert“ das Ideal <math>I</math>, es ist ein Annihilitor der Idealklassengruppe von <math>\mathbb{Q}(\zeta_p)</math>. Nach dem Satz von Stickelberger ist <math>S</math> ein Vielfaches des Stickelberger-Elements

:<math>\theta = \frac 1 p \sum_{a \in (\mathbb Z / p\mathbb Z)^*} a \sigma_a^{-1}.</math>

aus <math>\mathbb{Q}[G]</math>, das in <math> \mathbb{Z}[G]</math> liegt (das heißt, es gibt ein <math>\beta \in \mathbb{Z}[G]</math>, so dass <math>S = \beta\theta \in \mathbb{Z}[G]</math>).

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==References==
*Lothar Heffter, [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=249476 ''Ludwig Stickelberger''], Jahresbericht der [[German Mathematical Society|Deutschen Matematische Vereinigung]], XLVII (1937), pp.&nbsp;79–86

* Ludwig Stickelberger, [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27547 ''Ueber eine Verallgemeinerung der Kreistheilung''], Mathematische Annalen '''37''' (1890), pp.&nbsp;321–367

{{Authority control|VIAF=3243522}}

{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Stickelberger, Ludwig
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Swiss mathematician
| DATE OF BIRTH     = May 18, 1850
| PLACE OF BIRTH    = [[Buch, Schaffhausen]];
| DATE OF DEATH     = April 11, 1936
| PLACE OF DEATH    = [[Basel]]
}}
{{DEFAULTSORT:Stickelberger, Ludwig}}
[[Category:Swiss mathematicians]]
[[Category:Number theorists]]
[[Category:19th-century Swiss mathematicians]]
[[Category:1850 births]]
[[Category:1936 deaths]]
[[Category:ICM 1897 Plenary and Invited Speakers]]