Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a

Swiss Swiss may refer to: * the adjectival form of Switzerland * Swiss people Places * Swiss, Missouri * Swiss, North Carolina * Swiss, West Virginia *Swiss, Wisconsin Other uses * Swiss-system tournament, in various games and sports *Swiss Internatio ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History ...

who made important contributions to

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

(theory of

elementary divisors In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then ' ...

) and

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ...

(Stickelberger relation in the theory of

cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of t ...

s).

Short biography

Stickelberger was born in Buch in the

canton of Schaffhausen The canton of Schaffhausen, also canton of Schaffouse (german: Kanton Schaffhausen; rm, Chantun Schaffusa; french: Canton de Schaffhouse; it, Canton Sciaffusa) is the northernmost canton of Switzerland. The principal city and capital of the c ...

into a family of a pastor. He graduated from a gymnasium in 1867 and studied next in the

University of Heidelberg } Heidelberg University, officially the Ruprecht Karl University of Heidelberg, (german: Ruprecht-Karls-Universität Heidelberg; la, Universitas Ruperto Carola Heidelbergensis) is a public research university in Heidelberg, Baden-Württemberg, ...

. In 1874 he received a doctorate in Berlin under the direction of

Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

for his work on the transformation of

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

s to a diagonal form. In the same year, he obtained his

Habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...

from Polytechnicum in Zurich (now

ETH Zurich (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , aca ...

). In 1879 he became an extraordinary professor in the

Albert Ludwigs University of Freiburg The University of Freiburg (colloquially german: Uni Freiburg), officially the Albert Ludwig University of Freiburg (german: Albert-Ludwigs-Universität Freiburg), is a public research university located in Freiburg im Breisgau, Baden-Württe ...

. 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 11 April 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 Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...

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 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

Darboux Darboux is a surname. Notable people with the surname include: * Jean Gaston Darboux (1842–1917), French mathematician * Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021 * Paul Darboux (1919–1982), ...

. Augmented with the contemporaneous work of Frobenius, it set the theory of

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

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

that had just been developed by

Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...

Number theory

Three joint papers with Frobenius deal with the theory of

elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...

s. Today Stickelberger's name is most closely associated with his 1890 paper that established the

Stickelberger relation In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the ge ...

for cyclotomic

Gaussian sum In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a ...

s. This generalized earlier work of Jacobi and

Kummer Kummer is a German surname. Notable people with the surname include: * Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873—1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chri ...

and was later used by

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

in his formulation of the reciprocity laws in

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

s. The Stickelberger relation also yields information about the structure of the

class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a ...

of a

as a module over its abelian

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

(cf

Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the ea ...

References

*Lothar Heffter
Stickelberger''
Jahresbericht der Deutschen Matematische Vereinigung, XLVII (1937), pp. 79–86 * Ludwig Stickelberger
''Ueber eine Verallgemeinerung der Kreistheilung''
Mathematische Annalen 37 (1890), pp. 321–367

External links

* {{DEFAULTSORT:Stickelberger, Ludwig Swiss mathematicians Number theorists 19th-century Swiss mathematicians 1850 births 1936 deaths