Leopold Kronecker was a
GermanGermany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked on number theory and algebra.
He criticized Cantor's work on set theory, and was quoted by as having said, "
God made integers; all else is the work of man". Kronecker was a student and lifelong friend of
Ernst KummerErnst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
.
Biography
Leopold Kronecker was born in Liegnitz,
PrussiaPrussia was a German kingdom and historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organized and effective army. Prussia shaped the history...
(now Legnica,
PolandPoland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
) into a Jewish family. In 1845, Kronecker wrote his dissertation at the University of Berlin on
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, giving special formulation to units in certain algebraic number fields. Peter Gustav Dirichlet was his teacher.
After obtaining his degree, Kronecker managed the estate and business of his uncle, producing nothing mathematical for eight years. In his 1853 memoir on the algebraic solvability of equations, Kronecker extended the work of
Évariste GaloisÉvariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
on the
theory of equationsIn mathematics, the theory of equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.From the point...
. He accepted a professorship at
Friedrich-Wilhelms University (today: Humboldt University) of Berlin in 1883.
Kronecker also contributed to the concept of continuity, reconstructing the form of
irrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s in
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. In
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague,
Karl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
. In an 1850 paper,
On the Solution of the General Equation of the Fifth Degree, Kronecker solved the
quintic equationIn mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
by applying
group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
(though his solution was not in terms of radicals, since this was already proven impossible by
Abel–Ruffini theoremIn algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...
).
Kronecker's
finitismIn the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
made him a forerunner of
intuitionismIn the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
in
foundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
.
Named for Kronecker are the
Kronecker limit formulaIn mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series...
,
Kronecker delta,
Kronecker symbolIn number theory, the Kronecker symbol, written as \left or , is a generalization of the Jacobi symbol to all integers n. It was introduced by Leopold Kronecker.-Definition:...
,
Kronecker productIn mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...
,
Kronecker–Weber theoremIn algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
, Kronecker's method for factorizing polynomials,
Kronecker's theoremIn mathematics, Kronecker's theorem is either of two theorems named after Leopold Kronecker.- The existence of extension fields :This is a theorem stating that a polynomial in a field, p ∈ F[x], has a root in an extension field E \supset F.For example, a polynomial in the reals such...
in number theory, and
Kronecker's lemmaIn mathematics, Kronecker's lemma is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers...
. He was the supervisor of
Kurt HenselKurt Wilhelm Sebastian Hensel was a German mathematician born in Königsberg, Prussia.He was the son of the landowner and entrepreneur Sebastian Hensel, brother of the philosopher Paul Hensel, grandson of the composer Fanny Mendelssohn and the painter Wilhelm Hensel, and a descendant of the...
,
Adolf KneserAdolf Kneser was a German mathematician.He was born in Grüssow, Mecklenburg, Germany and died in Breslau, Germany ....
,
Mathias LerchMathias Lerch was an eminent Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory...
, and
Franz MertensFranz Mertens was a German mathematician. He was born in Środa in the Grand Duchy of Poznań, Kingdom of Prussia and died in Vienna, Austria....
, amongst others.
Kronecker died on December 29, 1891 in
BerlinBerlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
. He is buried in the
Alter St Matthäus Kirchhof Cemetery in
BerlinBerlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
-
SchönebergSchöneberg is a locality of Berlin, Germany. Until Berlin's 2001 administrative reform it was a separate borough including the locality of Friedenau. Together with the former borough of Tempelhof it is now part of the new borough of Tempelhof-Schöneberg....
, close to
Gustav KirchhoffGustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...
.
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