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 Internation ...
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
On ...
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 ob ...
(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 th ...
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
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
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 to a ...
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
, ac ...
). 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-Württemb ...
. 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
Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician.
Life
According this birth certificate he was bor ...
. Augmented with the contemporaneous work of Frobenius, it set the 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 '' ...
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 ...
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 ...
s. This generalized earlier work of
Jacobi Jacobi may refer to:
* People with the surname Jacobi (surname), Jacobi
Mathematics:
* Jacobi sum, a type of character sum
* Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
* Jacobi eigenva ...
and
Kummer 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 ...
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 mea ...
of a
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 th ...
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 pol ...
(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