Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a
German mathematician who worked on
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
and
logic, and criticized
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
's work on
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
.
Heinrich Weber quoted Kronecker
as having said, "'" ("God made the integers, all else is the work of man").
[The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886".]
Weber, Heinrich L. 1891–1892
Kronecker
''Jahresbericht der Deutschen Mathematiker-Vereinigung''
2:5-23. (The quote is on p. 19.)
Kronecker was a student and life-long friend of
Ernst Kummer.
Biography
Leopold Kronecker was born on 7 December 1823 in
Liegnitz,
Prussia (now Legnica,
Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
) in a wealthy
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
family. His parents, Isidor and Johanna (née Prausnitzep), took care of their children's education and provided them with private tutoring at home—Leopold's younger brother
Hugo Kronecker would also follow a scientific path, later becoming a notable
physiologist. Kronecker then went to the Liegnitz
Gymnasium where he was interested in a wide range of topics including
science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
,
history
History is the systematic study of the past, focusing primarily on the Human history, human past. As an academic discipline, it analyses and interprets evidence to construct narratives about what happened and explain why it happened. Some t ...
and
philosophy
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
, while also practicing gymnastics and swimming. At the gymnasium he was taught by
Ernst Kummer, who noticed and encouraged the boy's interest in mathematics.
In 1841 Kronecker became a student at the
University of Berlin where his interest did not immediately focus on mathematics, but rather spread over several subjects including
astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
and philosophy. He spent the summer of 1843 at the
University of Bonn studying astronomy and 1843–44 at the
University of Breslau following his former teacher Kummer. Back in Berlin, Kronecker studied mathematics with
Peter Gustav Lejeune Dirichlet and in 1845 defended his dissertation in
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 ...
written under Dirichlet's supervision.
After obtaining his degree, Kronecker did not follow his interest in research on an academic career path. He went back to his hometown to manage a large farming estate built up by his mother's uncle, a former banker. In 1848 he married his cousin Fanny Prausnitzer, and the couple had six children. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results.
In 1853 he wrote a memoir on the algebraic solvability of equations extending the work of
Évariste Galois on the
theory of equations.

Due to his business activity, Kronecker was financially comfortable, and thus he could return to Berlin in 1855 to pursue mathematics as a private scholar. Dirichlet, whose wife
Rebecka came from the wealthy
Mendelssohn family, had introduced Kronecker to the Berlin elite. He became a close friend of
Karl Weierstrass, who had recently joined the university, and his former teacher Kummer who had just taken over Dirichlet's mathematics chair.
[ Over the following years Kronecker published numerous papers resulting from his previous years' independent research. As a result of this published research, he was elected a member of the Berlin Academy in 1861.
Although he held no official university position, Kronecker had the right as a member of the Academy to hold classes at the University of Berlin and he decided to do so, starting in 1862. In 1866, when Riemann died, Kronecker was offered the mathematics chair at the University of Göttingen (previously held by Carl Friedrich Gauss and Dirichlet), but he refused, preferring to keep his position at the Academy. Only in 1883, when Kummer retired from the university, was Kronecker invited to succeed him and became an ordinary professor.] Kronecker was the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, and Franz Mertens, amongst others.
His philosophical view of mathematics put him in conflict with several mathematicians over the years, notably straining his relationship with Weierstrass, who almost decided to leave the university in 1888. Kronecker died on 29 December 1891 in Berlin, several months after the death of his wife. In the last year of his life, he converted to Christianity
Christianity is an Abrahamic monotheistic religion, which states that Jesus in Christianity, Jesus is the Son of God (Christianity), Son of God and Resurrection of Jesus, rose from the dead after his Crucifixion of Jesus, crucifixion, whose ...
.[ He is buried in the Alter St Matthäus Kirchhof cemetery in ]Berlin
Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...
- Schöneberg, close to Gustav Kirchhoff.
Scientific activity
Mathematics research
An important part of Kronecker's research focused on number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and algebra. In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker–Weber theorem, without however offering a definitive proof (the theorem was proved completely much later by David Hilbert). He also introduced the structure theorem for finitely-generated abelian groups. Kronecker studied elliptic functions and conjectured his " liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem. In an 1850 paper, ''On the Solution of the General Equation of the Fifth Degree'', Kronecker solved the quintic equation by applying group theory (though his solution was not in terms of radicals: that was already proven impossible by the Abel–Ruffini theorem).
In 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 ...
Kronecker introduced the theory of divisors as an alternative to Dedekind's theory of ideals, which he did not find acceptable for philosophical reasons. Although the general adoption of Dedekind's approach led Kronecker's theory to be ignored for a long time, his divisors were found useful and were revived by several mathematicians in the 20th century.
Kronecker also contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers. In analysis, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague, Karl Weierstrass.
Also named for Kronecker are the Kronecker limit formula, Kronecker's congruence, Kronecker delta, Kronecker comb, Kronecker symbol, Kronecker product, Kronecker's method for factorizing polynomials, Kronecker substitution, Kronecker's theorem in number theory, Kronecker's lemma, and Eisenstein–Kronecker numbers.
Philosophy of mathematics
Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
.
Honors
Kronecker was elected as a member of several academies:[
* Prussian Academy of Sciences (1861)
* ]French Academy of Sciences
The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...
(1868)
* Royal Society (1884).
The 25624 Kronecker asteroid is named after him.
Publications
*
*
References
Further reading
*
*
*
*
*
*
*
External links
*
*
{{DEFAULTSORT:Kronecker, Leopold
1823 births
1891 deaths
19th-century German mathematicians
Converts to Christianity
Corresponding members of the Saint Petersburg Academy of Sciences
Foreign members of the Royal Society
19th-century German Jews
Humboldt University of Berlin alumni
Academic staff of the Humboldt University of Berlin
Independent scientists
German number theorists
Intuitionism
People from Legnica
People from the Province of Silesia
Mathematicians from the Kingdom of Prussia