# Mathematician:Leopold Kronecker

## Contents

## Mathematician

German mathematician most notable for his view that all of mathematics ought to be based on integers.

Also a proponent of the mathematical philosophy of finitism, a forerunner of intuitionism and constructivism.

His influence on the mathematical establishment was considerable.

His views put him in direct opposition most notably to Georg Cantor, who was exploring the mathematics of the transfinite.

Solved the quintic equation using group theory, but this proof of course did not use radicals as this had been proved to be impossible by the Abel-Ruffini Theorem.

The first to provide an axiomatic formulation for the structure of an abstract group.

## Nationality

Prussian, then German

## History

- Born: December 7, 1823, Liegnitz, Prussia (now Legnica, Poland)
- Studied under Kummer
- 1841: Became a student at Berlin University, studied under Dirichlet and Steiner
- Summer of 1843: University of Bonn
- winter semester of 1843-44: University of Breslau, studied under Kummer again
- 1845: Submitted PhD thesis
- Left academia, returned home to Liegnitz to attend to family business
- 1848: Married Fanny Prausnitzer
- 1855: Returned to Berlin independently wealthy, but did not hold any university appointment
- 23 January 1861: Elected to the Berlin Academy
- October 1862: Started lecturing at Berlin University
- 1880: Took over control of
*Crelle's Journal*as the editor - 1883: Became a codirector of the mathematical seminar at Berlin University (taking over from Kummer)
- 31 January 1884: Elected a foreign member of the Royal Society of London
- Died: December 29, 1891, Berlin, Germany

## Theorems and Definitions

- First Kronecker Limit Formula
- Second Kronecker Limit Formula
- Kronecker Delta
- Kronecker Symbol
- Kronecker Sum
- Kronecker Product
- Kronecker-Weber Theorem
- Kronecker's Theorem in number theory
- Kronecker’s Theorem in field theory
- Kronecker's Lemma

- Kronecker-Capelli Theorem (with Alfredo Capelli) as it is known in Russia; also known as:

- Rouché-Fontené Theorem (after Eugène Rouché and Georges Fontené; priority actually goes to Georges Fontené) as it is known in France
- Rouché-Frobenius Theorem (after Eugène Rouché and Ferdinand Georg Frobenius)
- Rouché-Capelli Theorem (after Eugène Rouché and Alfredo Capelli) as it is known in Italy

- Kronecker's Constant, also known as:

Results named for **Leopold Kronecker** can be found here.

Definitions of concepts named for **Leopold Kronecker** can be found here.

## Publications

- 30 July 1845:
*De Unitatibus Complexis*("On complex units") (thesis) - 1850:
*On the Solution of the General Equation of the Fifth Degree* - 1887:
*Über den Zahlbergriff*

## Notable Quotes

*Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.*(translated loosely as:*God made the integers; all else is the work of man.*)- -- Quoted in:
- 1937: Eric Temple Bell:
*Men of Mathematics*:*They Say: What Say They? : Let Them Say* - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.): Introduction, also in section on $1$

- 1937: Eric Temple Bell:

- -- Quoted in:

## Requoted

*Plato said, "God is a geometer." Jacobi changed this to, "God is an arithmetician." Then came Kronecker and fashioned the memorable expression, "God created the natural numbers, and all the rest is the work of man."*- -- Felix Christian Klein
- -- Quoted as an epigraph to the preface to 1980: David M. Burton:
*Elementary Number Theory*(revised ed.)

*If, as Kronecker asserted, the integers are made by God and all the rest is the work of Man, then complex numbers are certainly one of Man's most intriguing mathematical artefacts.*- -- Opening sentence of 1983: Ian Stewart and David Tall:
*Complex Analysis (The Hitchhiker's Guide to the Plane)*

- -- Opening sentence of 1983: Ian Stewart and David Tall:

## Sources

- John J. O'Connor and Edmund F. Robertson: "Leopold Kronecker": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next):*They Say: What Say They? : Let Them Say* - 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{XXV}$ - 1983: Ian Stewart and David Tall:
*Complex Analysis (The Hitchhiker's Guide to the Plane)*... (next): $0$ The origins of complex analysis, and a modern viewpoint - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Introduction - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Introduction - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Kronecker, Leopold**(1823-91) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Kronecker, Leopold**(1823-91) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Kronecker, Leopold**(1823-91)