Search Results for Kronecker
Biographies
- Kronecker biography
- Leopold Kronecker
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- Leopold Kronecker's parents were well off, his father, Isidor Kronecker, being a successful business man while his mother was Johanna Prausnitzer who also came from a wealthy family.
- 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 Evangelical 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.
- Dirichlet commented on the thesis saying that in it Kronecker showed:-
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- 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.
- However, just as it looked as if he would embark on an academic career, Kronecker left Berlin to deal with family affairs.
- 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.
- Kronecker did not attract great numbers of students.
- Berlin was attractive to Kronecker, so much so that when he was offered the chair of mathematics in Gottingen 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 is well known for his remark:-
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- 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 Uber den Zahlbergriff in 1887.
- Another feature of Kronecker's personality was that he tended to fall out personally with those who 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.
- An example of how Kronecker reacted occurred in 1885 when Schwarz sent him a greeting which included the sentence:-
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- 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).
- Others however displayed more tact and, for example, Helmholtz who was a professor in Berlin from 1871, managed to stay on good terms with Kronecker.
- 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.
- However, Kronecker never addressed the meeting, since his wife was seriously injured in a climbing accident in the summer and died on 23 August 1891.
- Kronecker only outlived his wife by a few months, and died in December 1891.
- We should not think that Kronecker's views of mathematics were totally eccentric.
- Kronecker's ideas were further developed by Poincare and Brouwer, who placed particular emphasis upon intuition.
- Honours awarded to Leopold Kronecker
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- http://www-history.mcs.st-andrews.ac.uk/Biographies/Kronecker.html
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- Cantor biography
- Cantor attended lectures by Weierstrass, Kummer and Kronecker.
- A major paper on dimension which Cantor submitted to Crelle's Journal in 1877 was treated with suspicion by Kronecker, and only published after Dedekind intervened on Cantor's behalf.
- Cantor greatly resented Kronecker's opposition to his work and never submitted any further papers to Crelle's Journal.
- At one time it was thought that his depression was caused by mathematical worries and as a result of difficulties of his relationship with Kronecker in particular.
- he took a holiday in his favourite Harz mountains and for some reason decided to try to reconcile himself with Kronecker.
- Kronecker accepted the gesture, but it must have been difficult for both of them to forget their enmities and the philosophical disagreements between them remained unaffected.
- Cantor chaired the first meeting of the Association in Halle in September 1891, and despite the bitter antagonism between himself and Kronecker, Cantor invited Kronecker to address the first meeting.
- Kronecker never addressed the meeting, however, since his wife was seriously injured in a climbing accident in the late summer and died shortly afterwards.
- Hensel biography
- Among his teachers were Lipschitz, Weierstrass, Borchardt, Kirchhoff, Helmholtz and Kronecker.
- It was Kronecker who was the greatest influence on Hensel and supervised his doctoral studies at the University of Berlin.
- He devoted many years to the editing of Kronecker's collected works.
- In fact he published five volumes of Kronecker's works between the years 1895 and 1930.
- Two other major volumes edited by Hensel were also devoted to Kronecker's works.
- Hensel's work followed that of his doctoral supervisor Kronecker in the development of arithmetic in algebraic number fields.
- This fact had already been pointed out in articles of Kronecker (who supervised Hensel's doctorate) and of Dedekind and Heinrich Weber, which had been published in 1881 and 1882, respectively, the paper of Kronecker based on a then unpublished manuscript from the year 1858.
- Eisenstein biography
- Writing of his mathematical works written during this period Weil writes in [Elliptic functions according to Eisenstein and Kronecker (Berlin, 1999).',3)">3]:-
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- Weil writes in [Elliptic functions according to Eisenstein and Kronecker (Berlin, 1999).',3)">3]:-
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- Kronecker wrote (see for example [Elliptic functions according to Eisenstein and Kronecker (Berlin, 1999).',3)">3]):-
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- In fact the book [Elliptic functions according to Eisenstein and Kronecker (Berlin, 1999).',3)">3], the first edition of which appeared in 1976 and was the result of a course given at the Institute for Advanced Study at Princeton in 1974, is devoted to this approach.
- Kronecker took up these themes [Elliptic functions according to Eisenstein and Kronecker (Berlin, 1999).',3)">3]:-
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- much of Kronecker's best work consists of such variations ..
- Kummer biography
- His two most famous pupils were Kronecker and Joachimsthal and, under Kummer's guidance, they were undertaking mathematical research while at school.
- Kronecker had also been appointed to Berlin in 1855 so Berlin became one of the leading mathematical centres in the world.
- While Weierstrass and Kronecker offered the most recent results of their research in their lectures, Kummer in his restricted himself, after instituting the seminar, to laying firm foundations.
- The three great mathematicians of Berlin, Kummer, Weierstrass and Kronecker were close friends for twenty years as they worked closely and effectively together, However, around 1875 Weierstrass and Kronecker fell out.
- Kummer continued his friendship with Kronecker but this put a strain on his relation with Weierstrass.
- Perhaps it is not too surprising that this should happen to these three great mathematical talents, particularly given that Kronecker vigorously attacked personally anyone with whom he had a mathematical difference.
- Castelnuovo biography
- Castelnuovo is also remembered for the Kronecker-Castelnuovo theorem which states:-
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- Kronecker had first stated a version of this theorem in a lecture which he gave to the Accademia dei Lincei in 1886.
- Castelnuovo had only recently graduated when he was informed by Cremona of Kronecker's lecture and he found his own proof of the result.
- Kronecker never published the theorem and it was Castelnuovo's version which appeared in print.
- Frobenius biography
- Back at the University of Berlin he attended lectures by Kronecker, Kummer and Weierstrass.
- In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant.
- That he could not prevent this, that he could not reach his goal of maintaining unchanged the times of Weierstrass, Kummer and Kronecker also in their external appearances, but to witness helplessly these developments, was doubly intolerable for him, with his choleric disposition.
- In the introduction to this paper he explains how he became interested in abstract groups, and this was through a study of one of Kronecker's papers.
- Holder biography
- He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series.
- At Berlin he was a fellow student of Runge and he attended lectures by Weierstrass, Kronecker and Kummer.
- Holder's interest in algebra came partly through the influence of Kronecker at this time and Kronecker's liking for rigour almost certainly was to have a profound influence on Holder's later work in algebra.
- Bauer Mihaly biography
- These were Uber einen Satz von Kronecker and Uber zusammengesetzte Korper.
- These papers made an important contribution to Kronecker's question concerning characterising number fields by the splitting behaviour of their primes.
- Kronecker called this a 'boundary value problem' (Randwertproblem) because of a (vague) analogy with Cauchy's theorem computing the values of an analytic function on a disc from its values taken at the boundary.
- In the second of the papers he completely solved Kronecker's question concerning characterising number fields by the splitting behaviour of their primes.
- Casorati biography
- On this and later journeys, for example he discussed the foundations of analysis with Kronecker and Weierstrass.
- Among the mathematicians with whom Casorati corresponded we find Enestrom, Fuchs, Hermite, Kronecker, Schwarz, Prym, Mittag-Leffler and Schlafli (no fewer than 108 letters with him).
- Conversations with Kronecker and Weierstrass are reported.
- Weierstrass biography
- At Berlin, Weierstrass had two colleagues Kummer and Kronecker and together the three gave Berlin a reputation as the leading university at which to study mathematics.
- Kronecker was a close friend of Weierstrass's for many years but in 1877 Kronecker's opposition to Cantor's work cause a rift between the two men.
- Konig Julius biography
- After being awarded a doctorate from Heidelberg Konig went to Berlin where he spent six months attending lectures by Weierstrass and Kronecker.
- His most important work written in 1903 is based on a fundamental study by Kronecker published in 1892.
- Konig developed Kronecker's polynomial ideals and presented many results on discriminants of forms, elimination theory and Diophantine problems.
- Maschke biography
- At Berlin he was taught by some outstanding mathematical teachers including Weierstrass, Kummer and Kronecker.
- Hermite, Kronecker and Brioschi had, in 1858, discovered how to solve the quintic equation by means of elliptic functions.
- Among the papers he published while at Chicago are: On systems of six points lying in three ways in involution (1896), Note on the unilateral surface of Mobius (1900), A new method of determining the differential parameters and invariants of quadratic differential quantics (1900), On superosculating quadric surfaces (1902), A symbolic treatment of the theory of invariants of quadratic differential quantics of n variables (1903), Differential parameters of the first order (1906); The Kronecker-Gaussian curvature of hyperspace (1906) and A geometrical problem connected with the continuation of a power-series (1906).
- Gutzmer biography
- Among those who lectured to Gutzmer in mathematics were L Kronecker, K Weierstrass and L Fuchs.
- Kronecker died in 1891 and Schwarz succeeded Weierstrass one year later.
- Gutzmer, therefore studied with two of the three great Berlin mathematicians, Kronecker and Weierstrass, in the last years of their careers.
- Hecke biography
- Complex multiplication and modular forms had been treated in the 19th century by Kronecker and Heinrich Weber, who discovered their link with class field theory.
- For his doctoral work Hilbert suggested to Hecke that he extend Kronecker's ideas to curves of genus 2.
- Hunyadi biography
- Perhaps the most fruitful of all the visits he made was to Berlin where the remarkable team of Kummer, Borchardt, Weierstrass and Kronecker were lecturing.
- It was the lecture courses given by Kummer and Kronecker which had the greatest influence on Hunyadi.
- Christoffel biography
- He and his colleague Aronhold tried to attract high quality students to the Gewerbsakademie but this proved difficult with the highly prestigious University of Berlin with Weierstrass, Kummer and Kronecker close by.
- In our opinion Christoffel's teacher Dirichlet, belongs to the next most important group of mathematicians which includes (in chronological order of birth) Jacobi, Kummer, Kronecker, Dedekind, Cantor and Klein.
- Fine Henry biography
- Fine spent the summer of 1885 in Berlin attending Kronecker's lectures on eliminants which made a strong impression on him.
- He gave his retiring address as president on An unpublished theorem of Kronecker respecting numerical equations.
- Hurwitz biography
- Although he was greatly influenced by Klein and had already begun to undertake advanced work with him, he went for academic year 1877-78 to continue his studies at the University of Berlin where he attended classes by Kummer, Weierstrass and Kronecker.
- Hurwitz had not been at Munich during 1881-82, rather he had returned to Berlin where he attended further courses of lectures by Weierstrass and Kronecker.
- Takagi biography
- On his return to Tokyo in 1903 Takagi proved a conjecture on abelian extensions of imaginary number fields made by Kronecker.
- Kronecker described this conjecture as:-
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- Runge biography
- After qualifying to be a Gymnasium teacher during session 1880-81, he completed the necessary examinations and returned to Berlin where he began to collaborate with Kronecker.
- This entitled him to lecture at the University of Berlin, and there he continued to undertake research on algebra and function theory as part of the group of mathematicians which had built up around Kronecker.
- Nugel biography
- Until he last years of the 19th century, Berlin had been the leading centre for mathematics in Germany, perhaps in the world, with Kummer, Weierstrass and Kronecker.
- However Schwarz had succeeded Weierstrass accepting a professorship in Berlin in 1892; Kronecker died in 1891 and was succeeded by Frobenius; and Kummer had retired and been replaced by Fuchs.
- Mittag-Leffler biography
- Hardly ever was there such a brilliant collection of distinguished mathematicians: Weierstrass, Kummer, Kronecker, Helmholtz, Kirchhoff, Borchardt etc.
- Mittag-Leffler was one of the first mathematicians to support Cantor's theory of sets but, one has to remark, a consequence of this was that Kronecker refused to publish in Acta Mathematica .
- Kneser biography
- He was taught at the University of Berlin by Kronecker and also influenced by Weierstrass.
- Kronecker and Kummer supervised his doctoral studies which ended in 1884 with the award of the degree from Berlin for his thesis Irreduktibilitat und Monodromiegruppe algebraischer Gleichungen.
- Dedekind biography
- In Berlin, Dedekind met Weierstrass, Kummer, Borchardt and Kronecker.
- In this quote Dedekind is arguing against Kronecker's objections to the infinite and, therefore, is agreeing with Cantor's views.
- Mertens biography
- Mertens completed his university studies at the University of Berlin where he attended lectures by Weierstrass, Kronecker and Kummer.
- His advisors had been Kummer and Kronecker.
- Schlesinger biography
- His thesis advisors were Lazarus Immanuel Fuchs and Leopold Kronecker.
- Wiltheiss biography
- There he attended lectures by the three great mathematicians Weierstrass, Kummer, and Kronecker.
- Netto biography
- He again had some inspiring teachers in Kronecker, Weierstrass and Kummer.
- Vandermonde biography
- Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde.
- Wangerin biography
- His appointment was specifically to teach first year undergraduates, for neither Kronecker nor Weierstrass lectured at this level.
- Ribenboim biography
- The first of these is devoted to ramification theory in Galois extensions and the second to a proof of the theorem by Kronecker and Heinrich Weber on the abelian extensions of the field of rational numbers.
- Hasse biography
- He extended Heinrich Weber's work on class field theory writing several important papers and starting work on his famous report on class field theory which included the contributions of Kronecker, Heinrich Weber, Hilbert, Furtwangler and Takagi.
- Bromwich biography
- This book is an early example in English of the more abstract methods introduced into algebra by researchers such as Kronecker and Weierstrass.
- Valyi biography
- This scholarship enabled him to spend two years studying at the University of Berlin where the remarkable team of Kummer, Borchardt, Weierstrass and Kronecker were lecturing.
- Suter biography
- He then studied at Berlin under Kronecker, Kummer and Weierstrass.
- Henrici biography
- Henrici then went to Berlin and studied under Weierstrass and Kronecker.
- Roch biography
- Following his time in Gottingen, Roch went to Berlin where he made contact with L Kronecker, E E Kummer, K Weierstrass and K W Borchardt.
- Lerch biography
- After leaving Prague he went to the University of Berlin where he studied during 1884 -85 and was taught by Weierstrass, Kronecker and Fuchs.
- Couturat biography
- Dedekind, Kronecker, and Helmholtz were already strong advocates of formalist theories so Couturat took a stand against major established figures - a brave move in a doctoral thesis.
- Thue biography
- He also spent a while in Berlin where he attended lectures by Helmholtz, Fuchs and Kronecker.
- Meders biography
- Adolf Kneser, who had been taught by Kronecker and written a thesis on algebraic functions and equations, was the professor at Dorpat.
- Lie biography
- In Berlin he met Kronecker, Kummer and Weierstrass.
- Pasch biography
- There he studied under Weierstrass and Kronecker.
- Weil biography
- Weil's most famous books include Foundations of Algebraic Geometry (1946) and Elliptic Functions According to Eisenstein and Kronecker (1976).
- Sylow biography
- In Berlin he had useful discussions with Kronecker but was unable to attend courses by Weierstrass who was ill at the time.
- Chebyshev biography
- In addition to the mathematicians we have mentioned that he met on that trip, he also had contacts with other European mathematicians such as Lucas, Borchardt, Kronecker, and Weierstrass (see for example [J.
- Weyr Eduard biography
- In 1885-6 he took lectures of Kronecker and Weierstrass in Berlin.
- Motzkin biography
- He continued to study mathematics through university being accepted as a research student by Kronecker.
- Klein biography
- After building on methods due to Hermite and Kronecker, producing similar results to Brioschi, he went on to completely solve the problem using the group of the icosahedron.
- White biography
- White's friend, William J James, had studied under Klein at Gottingen and also under Kronecker and Fuchs in Berlin.
- Vandiver biography
- He was not, however, the first to make the conjecture which should really be named 'Kummer's conjecture' since it first appears in 1849 in a letter which Kummer wrote to Kronecker.
- Bolza biography
- He returned to the University of Berlin where he worked with Kronecker and Fuchs but, after corresponding with Klein about the results he was obtaining, he decided to make one final change in his long route to a doctorate and went to Gottingen to be supervised by Klein.
- Gibson biography
- This was the period when the University of Berlin was at its height with Weierstrass, Kummer and Kronecker all giving outstanding lecture courses.
- Mackey biography
- He then produced a series of important papers on group representations including On induced representations of groups (1951), Induced representations of locally compact groups (1952), and Symmetric and anti symmetric Kronecker squares and intertwining numbers of induced representations of finite groups (1953).
- Hoehnke biography
- The main directions of this theory should be: the theory of (0)-primative semigroups, the 0-radical, simplifications of the theory under finiteness conditions, a study of the congruence lattice, Kronecker products (coming from matrix theory) for semigroups and acts, topological methods, comparison of the lift- and right-sided behaviour of a semigroup and finally, finding the place of all former results ..
- Plessner biography
- Then he worked in Marburg with Hensel editing Kronecker's collected works.
- Heine biography
- However, he also got to know the other outstanding mathematicians in Berlin: Weierstrass, Kummer, Kronecker, and Borchardt.
- Meyer biography
- He attended the University of Leipzig and studied at the University of Berlin taking courses by Kummer, Weierstrass and Kronecker.
- Hilbert biography
- The Zahlbericht (1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas.
- Stolz biography
- He went first to Berlin where he attended lectures by the three great mathematicians Weierstrass, Kummer and Kronecker.
- Moore Eliakim biography
- Moore spent the year in Germany, going first to Gottingen where he spent the summer of 1885 studying the German language, but spending most of the academic year 1885-86 attending lectures by Kronecker and Weierstrass at the University of Berlin.
- Brioschi biography
- This problem was also solved by Kronecker at almost exactly the same time.
- Herbrand biography
- These papers simplify proofs of results by Kronecker, Heinrich Weber, Hilbert, Takagi and Artin.
- Nassau biography
- Finally we list a few of Nassau's papers: Questions and Discussions: Discussions: Evaluation of the Determinant |1/(r + s - 1)! | (1924); Some extensions of the generalized Kronecker symbol (1926); Questions and Discussions: Discussions: Concerning a Theorem in Determinants (1927); and (with O E Brown) A Navigation Computer (1947).
- Koksma biography
- The approximation theorem of Kronecker is discussed at length.
- Hankel biography
- From Leipzig he went to Gottingen in 1860 where he became a student of Riemann and then, in the following year, he worked with Weierstrass and Kronecker in Berlin.
- Gegenbauer biography
- He then went to Berlin where he studied from 1873 to 1875 working under Weierstrass and Kronecker.
- Dirichlet biography
- Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties.
- Heisenberg biography
- In fact by this time he had become interested in number theory and he read Kronecker's work and tried to find a proof of Fermat's Last Theorem.
- Stackel biography
- Among his teachers were the mathematicians Kronecker, Kummer, Wangerin and Weierstrass, the latter seemingly having the biggest influence in the early years of his career.
History Topics
- Set theory
- Abstract groups
- In 1870 Kronecker gave a definition of a group in a completely different context, namely the context of a class group in algebraic number theory.
- This appears to be a separate development by Kronecker who does not tie it in with previous work on groups.
- However, Heinrich Weber in 1882 gave a very similar definition to that of Kronecker yet he did tie it in with previous work on groups.
- Heinrich Weber defined a group of degree h, like Kronecker in the context of class groups, again to be a finite set.
- Matrices and determinants
- Jacobi from around 1830 and then Kronecker and Weierstrass in the 1850's and 1860's also looked at matrix results but again in a special context, this time the notion of a linear transformation.
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- He cites Kronecker and Weierstrass as having considered special cases of his results in 1874 and 1868 respectively.
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- In the same year Kronecker's lectures on determinants were also published, again after his death.
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- Set theory references
- A Fuhrich, Der Meinungsstreit zwischen Georg Cantor und Leopold Kronecker um Grundlagen der Mathematik in der Zeit der Begrundung der Mengenlehre (Potsdam, 1983).
- Set theory references
- A Fuhrich, Der Meinungsstreit zwischen Georg Cantor und Leopold Kronecker um Grundlagen der Mathematik in der Zeit der Begrundung der Mengenlehre (Potsdam, 1983).
- Ring Theory
- This term, invented by Kronecker, is still used today in algebraic number theory.
- Group theory
Famous Curves
No matches from this section
Societies etc
- German Mathematical Society
- Despite the bitter antagonism that existed between Cantor and Kronecker, Cantor invited Kronecker to address this first meeting of the Society in 1891 as a sign of respect for one of the senior and most eminent figures in German mathematics.
- However, Kronecker never addressed the meeting, since his wife suffered a serious accident prior to the meeting.
- LMS Honorary Member
- Hungarian Academy of Sciences
- In mathematics Cayley, Hermite, Helmholtz, Kronecker, Du Bois-Reymond, Fuchs, Klein, Stackel, Darboux and Mittag-Leffler all became members.
- Fellow of the Royal Society
References
- References for Kronecker
- References for Leopold Kronecker
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- G Frobenius, Gedachtnisrede auf Leopold Kronecker, Abhandlungen der Koniglich Preussichen Akademie der Wissenschaften zu Berlin (1893).
- W Purkert, Kronecker, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
- F Arzarello, The finite in Kronecker (Italian), Epistemology of mathematics: 1989-1991 Seminars (Italian) (Rome, 1992), 135-146.
- R Crespo, Leopoldo Kronecker (Spanish), Gaceta Mat.
- H M Edwards, An Appreciation of Kronecker, The Mathematical Intelligencer 9 (1987), 28-35.
- H M Edwards, Kronecker's views on the foundations of mathematics, in The history of modern mathematics Vol.
- H M Edwards, Kronecker's place in history, in History and philosophy of modern mathematics (Minneapolis, MN, 1988), 139-144.
- H M Edwards, On the Kronecker Nachlass, Historia Math.
- H M Edwards, Kronecker's arithmetical theory of algebraic quantities, Jahresberichte der Deutschen Mathematiker-Vereinigung 94 (3) (1992), 130-139.
- F Gana, God and Man in Kronecker's mathematics (Italian), Historia Math.
- H Perfect, Leopold Kronecker : a great gentleman in science, Mathematical Spectrum 24, 1-7.
- H Weber, Leopold Kronecker, Jahresberichte der Deutschen Mathematiker-Vereinigung 2 (1893), 5-23.
- http://www-history.mcs.st-andrews.ac.uk/References/Kronecker.html
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- References for Weierstrass
- K-R Biermann, Kontroversen um den Steiner-Preis und ihre Folgen : Ein Kapitel aus den Beziehungen zwischen Weierstrass und Kronecker, Historia Sci.
- References for Eisenstein
- A Weil, Elliptic functions according to Eisenstein and Kronecker (Berlin, 1999).
- References for Dirichlet
- E E Kummer, Peter Gustav Lejeune Dirichlet, in L Kronecker and L Fuchs, G Lejeune Dirichlets Werke (Berlin, 1889-97).
Additional material
- Jacobson: 'Structure of Rings
- In Chapter V we define Kronecker products of modules and algebras and we reduce the problems of determining the structure of Kronecker products of simple algebras to the case of division algebras and fields.
- We consider the Galois theory of automorphisms for division rings, the structure of Kronecker products of division rings, and commutativity theorems (e.g.
- EMS obituary
- Another achievement of Segre's was to geometrise Kronecker's results on a pencil of singular quadratic forms.
- The difficulty of Kronecker's algebraic reduction is notorious but Segre, by equating a singular form to zero and interpreting the equation as a cone in [n], where n + 1 is the number of variables in the form, considers the locus of vertices (which need not be points merely, but spaces of larger dimension) of the cones.
- To the geometrically minded, to be able to state a geometrical criterion for all the quadrics of the pencil to be singular, of a specified degree, will appeal as a consummation of Kronecker's remarkable work.
- Turnbull and Aitken: 'Canonical Matrices
- The reader already familiar with the theory will also observe that certain established methods of dealing with the subject have hardly been touched upon, notably the methods of Weierstrass and Darboux, the theory of regular minors of determinants and the treatment of quadratic forms by the methods of Kronecker.
- Our tribute to Kronecker finds expression in Chapter IX, which is an essay towards giving a fresh derivation of his classical results concerning singular pencils; we have treated this by rational methods, and we trust that an intricate argument has been materially simplified.
- EMS obituary
- The proof leans heavily on work of Kronecker on how to define an algebraic construct by systems of equations; but it also uses defective integrals, and before considering Baker's later work as a geometer these should be described in some detail.
- Chapter VII, though perhaps too compressed for so intricate a matter, is concerned with work of Kronecker and Dedekind on the relation between everywhere finite integrals and integral functions.
- James Jeans addresses the British Association in 1934
- It was, I think, Kronecker who said that in arithmetic God made the integers and man made the rest; in the same spirit, we may add that in physics God made the mathematics and man made the rest.
- H F Baker: 'A locus with 25920 linear self-transformations' Introduction
- Later in the same volume, with acknowledgements to Kronecker, he considers the group of the trisection of the periods of a theta function of two variables, proving that the study of this group is essentially the same problem as that of the group of the lines of a cubic surface.
- Malcev: 'Foundations of Linear Algebra' Introduction
- Results which appeared near the end of the 19th century included the normal form of a matrix of a linear transformation (Jordan), elementary divisors (Weierstrass), pairs of quadratic forms (Weierstrass, Kronecker), and Hermitian forms (Hermite).
- R L Wilder: 'Cultural Basis of Mathematics III
- Consider, for example, the insistence of Intuitionism that all mathematics should be founded on the natural numbers or the counting process, and that the latter are "intuitively given." There are plausible arguments to support the thesis that the natural numbers should form the starting point for mathematics, but it is hard to understand just what "intuitively given" means, or why the classical conception of the continuum, which the Intuitionist refuses to accept, should not be considered as "intuitively given." It makes one feel that the Intuitionist has taken Kronecker's much-quoted dictum that "The integers were made by God, but all else is the work of man"' and substituted "Intuition" for "God." However, if he would substitute for this vague psychological notion of "intuition" the viewpoint that inasmuch as the counting process is a cultural invariant, it follows that the natural numbers.
- J Ruska on Heinrich Suter
- In Berlin he attended lectures by Kummer, Weierstrass and Kronecker.
- James Jeans: 'Physics and Philosophy' II
- Kronecker is quoted as saying that in arithmetic God made the integers and man made the rest; in the same spirit we may perhaps say that in physics God made the mathematics and man made the rest.
- David Hilbert: 'Mathematical Problems
- For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors - a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the centre of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.
- H F Baker's locus with 25920 linear self-transformations - Introduction
- Later in the same volume, with acknowledgements to Kronecker, he considers the group of the trisection of the periods of a theta function of two variables, proving that the study of this group is essentially the same problem as that of the group of the lines of a cubic surface.
- Percy MacMahon addresses the British Association in 1901, Part 2
- Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poincare, and Klein are great names that will be for ever associated with it.
Quotations
- Quotations by Kronecker
- Quotations by Leopold Kronecker
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- From Weber's obituary of Kronecker
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- http://www-history.mcs.st-andrews.ac.uk/Quotations/Kronecker.html
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Chronology
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JOC/BS August 2001