**Leopold Kronecker**'s primary contributions were in the theory of equations. He made major contributions in elliptic functions and the theory of algebraic numbers.

- The families were Jewish, the religion that Kronecker kept until a year before his death when he became a convert to Christianity.
- Kronecker's parents employed private tutors to teach him up to the stage when he entered the Gymnasium at Liegnitz, and this tutoring gave him a very sound foundation to his education.
- Kronecker was taught mathematics at Liegnitz Gymnasium by Kummer, and it was due to Kummer that Kronecker became interested in mathematics.
- Kummer immediately recognised Kronecker's talent for mathematics and he took him well beyond what would be expected at school, encouraging him to undertake research.
- Despite his Jewish upbringing, Kronecker was given Lutheran religious instruction at the Gymnasium which certainly shows that his parents were openminded on religious matters.
- Kronecker became a student at Berlin University in 1841 and there he studied under Dirichlet and Steiner.
- Kronecker spent a year at Breslau before returning to Berlin for the winter semester of 1844-45.
- It may come as a surprise to many Ph.D. students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
- Eisenstein, whose health was also poor, lectured in Berlin around this time and Kronecker came to know both men well.
- The direction that Kronecker's mathematical interests went later had much to do with the influence of Jacobi and Eisenstein around this time.
- Certainly Kronecker did not need to take on paid employment since he was by now a wealthy man.
- In 1856 Weierstrass came to Berlin, so within a year of Kronecker returning to Berlin, the remarkable team of Kummer, Borchardt, Weierstrass and Kronecker was in place in Berlin.
- Of course since Kronecker did not hold a university appointment, he did not lecture at this time but was remarkably active in research publishing a large number of works in quick succession.
- Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass.
- On 23 January 1861 Kronecker was elected to the Academy and this had a surprising benefit.
- Although Kronecker was not employed by the University, or any other organisation for that matter, Kummer suggested that Kronecker exercise his right to lecture at the University and this he did beginning in October 1862.
- Berlin was attractive to Kronecker, so much so that when he was offered the chair of mathematics in Göttingen in 1868, he declined.
- In order to understand why relations began to deteriorate in the 1870s we need to examine Kronecker's mathematical contributions more closely.
- We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.
- Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations.
- It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature.
- Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist.
- In 1870 Heine published a paper On trigonometric series in Crelle's Journal, but Kronecker had tried to persuade Heine to withdraw the paper.
- Again in 1877 Kronecker tried to prevent publication of Cantor's work in Crelle's Journal, not because of any personal feelings against Cantor (which has been suggested by some biographers of Cantor) but rather because Kronecker believed that Cantor's paper was meaningless, since it proved results about mathematical objects which Kronecker believed did not exist.
- Kronecker was on the editorial staff of Crelle's Journal which is why he had a particularly strong influence on what was published in that journal.
- After Borchardt died in 1880, Kronecker took over control of Crelle's Journal as the editor and his influence on which papers would be published increased.
- The mathematical seminar in Berlin had been jointly founded in 1861 by Kummer and Weierstrass and, when Kummer retired in 1883, Kronecker became a codirector of the seminar.
- This increased Kronecker's influence in Berlin.
- Kronecker's international fame also spread, and he was honoured by being elected a foreign member of the Royal Society of London on 31 January 1884.
- Although Kronecker's view of mathematics was well known to his colleagues throughout the 1870s and 1880s, it was not until 1886 that he made these views public.
- Lindemann had proved that π is transcendental in 1882, and in a lecture given in 1886 Kronecker complimented Lindemann on a beautiful proof but, he claimed, one that proved nothing since transcendental numbers did not exist.
- So Kronecker was consistent in his arguments and his beliefs, but many mathematicians, proud of their hard earned results, felt that Kronecker was attempting to change the course of mathematics and write their line of research out of future developments.
- Kronecker explained his programme based on studying only mathematical objects which could be constructed with a finite number of operation from the integers in Über den Zahlbergriff Ⓣ(On the number of terms) in 1887.
- Another feature of Kronecker's personality was that he tended to fall out personally with those whom he disagreed with mathematically.
- Not only Dedekind, Heine and Cantor's mathematics was unacceptable to this way of thinking, and Weierstrass also came to feel that Kronecker was trying to convince the next generation of mathematicians that Weierstrass's work on analysis was of no value.
- Kronecker had no official position at Berlin until Kummer retired in 1883 when he was appointed to the chair.
- But by 1888 Weierstrass felt that he could no longer work with Kronecker in Berlin and decided to go to Switzerland, but then, realising that Kronecker would be in a strong position to influence the choice of his successor, he decided to remain in Berlin.
- Kronecker was of very small stature and extremely self-conscious about his height.
- Here Schwarz was joking about the small man Kronecker and the large man Weierstrass.
- Kronecker did not see the funny side of the comment, however, and never had any further dealings with Schwarz (who was Weierstrass's student and Kummer's son-in-law).
- Despite the bitter antagonism between Cantor and Kronecker, Cantor invited Kronecker to address this first meeting as a sign of respect for one of the senior and most eminent figures in German mathematics.
- We should not think that Kronecker's views of mathematics were totally eccentric.
- Kronecker's ideas were further developed by Poincaré and Brouwer, who placed particular emphasis upon intuition.

Born 7 December 1823, Liegnitz, Prussia (now Legnica, Poland). Died 29 December 1891, Berlin, Germany.

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Algebra, Group Theory, Origin Poland, Set Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive