Leopold Kronecker | |
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 Kronecker in 1865 | |
| Born | (1823-12-07)7 December 1823 |
| Died | 29 December 1891(1891-12-29) (aged 68) |
| Alma mater | University of Berlin |
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| Awards | ForMemRS (1884) |
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Leopold Kronecker (German:[ˈkʁoːnɛkɐ]; 7 December 1823 – 29 December 1891) was aGermanmathematician who worked onnumber theory,abstract algebra andlogic, and criticizedGeorg Cantor's work onset theory. Heinrich Weber quoted Kronecker[1]as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man").[2]Kronecker was a student and life-long friend ofErnst Kummer.
Leopold Kronecker was born on 7 December 1823 inLiegnitz,Prussia (now Legnica,Poland) in a wealthyJewish family. His parents, Isidor and Johanna (née Prausnitzer), took care of their children's education and provided them with private tutoring at home—Leopold's younger brotherHugo Kronecker would also follow a scientific path, later becoming a notablephysiologist. Kronecker then went to the LiegnitzGymnasium where he was interested in a wide range of topics includingscience,history andphilosophy, while also practicing gymnastics and swimming. At the gymnasium he was taught byErnst Kummer, who noticed and encouraged the boy's interest in mathematics.[3]
In 1841 Kronecker became a student at theUniversity of Berlin where his interest did not immediately focus on mathematics, but rather spread over several subjects includingastronomy and philosophy. He spent the summer of 1843 at theUniversity of Bonn studying astronomy and 1843–44 at theUniversity of Breslau following his former teacher Kummer. Back in Berlin, Kronecker studied mathematics withPeter Gustav Lejeune Dirichlet and in 1845 defended his dissertation inalgebraic number theory written under Dirichlet's supervision.[4]
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.[5] In 1853 he wrote a memoir on the algebraic solvability of equations extending the work ofÉvariste Galois on thetheory 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 wifeRebecka came from the wealthyMendelssohn family, had introduced Kronecker to the Berlin elite. He became a close friend ofKarl Weierstrass, who had recently joined the university, and his former teacher Kummer who had just taken over Dirichlet's mathematics chair.[3] 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 theBerlin 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, whenRiemann died, Kronecker was offered the mathematics chair at theUniversity of Göttingen (previously held byCarl 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.[6] Kronecker was the supervisor ofKurt Hensel,Adolf Kneser,Mathias Lerch, andFranz 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.[4] 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 toChristianity.[3] He is buried in theAlter St Matthäus Kirchhof cemetery inBerlin-Schöneberg, close toGustav Kirchhoff.
An important part of Kronecker's research focused onnumber theory andalgebra. In an 1853 paper on thetheory of equations andGalois theory he formulated theKronecker–Weber theorem, without however offering a definitive proof (the theorem was proved completely much later byDavid Hilbert). He also introduced thestructure theorem for finitely generated abelian groups. Kronecker studiedelliptic functions and conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as histwelfth problem.[7] In an 1850 paper,On the Solution of the General Equation of the Fifth Degree, Kronecker solved thequintic equation by applyinggroup theory (though his solution was not in terms of radicals: that was already proven impossible by theAbel–Ruffini theorem).
Inalgebraic number theory Kronecker introduced thetheory of divisors as an alternative toDedekind's theory ofideals, 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.[8]
Kronecker also contributed to the concept ofcontinuity, reconstructing the form ofirrational numbers inreal numbers. Inanalysis, Kronecker rejected the formulation of acontinuous, nowhere differentiable function by his colleague,Karl Weierstrass.
Also named for Kronecker are theKronecker 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, andEisenstein–Kronecker numbers.
Kronecker'sfinitism made him a forerunner ofintuitionism infoundations of mathematics.
Kronecker was elected as a member of several academies:[3]
The25624 Kronecker asteroid is named after him.
