Leopold Kronecker
Scientific activity
Scientific activity
-------------------

### Mathematics research

An important part of Kronecker's research focused on [number theory](/wiki/Number_theory "Number theory") and [algebra](/wiki/Algebra "Algebra"). In an 1853 paper on the [theory of equations](/wiki/Theory_of_equations "Theory of equations") and [Galois theory](/wiki/Galois_theory "Galois theory") he formulated the [Kronecker–Weber theorem](/wiki/Kronecker%E2%80%93Weber_theorem "Kronecker–Weber theorem"), without however offering a definitive proof (the theorem was proved completely much later by [David Hilbert](/wiki/David_Hilbert "David Hilbert")). He also introduced the [structure theorem for finitely\-generated abelian groups](/wiki/Finitely-generated_abelian_group "Finitely-generated abelian group"). Kronecker studied [elliptic functions](/wiki/Elliptic_functions "Elliptic functions") and conjectured his "[liebster Jugendtraum](/wiki/Kronecker%27s_Jugendtraum "Kronecker's Jugendtraum")" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his [twelfth problem](/wiki/Hilbert%27s_problems "Hilbert's problems"). In an 1850 paper, *On the Solution of the General Equation of the Fifth Degree*, Kronecker solved the [quintic equation](/wiki/Quintic_equation "Quintic equation") by applying [group theory](/wiki/Group_theory "Group theory") (though his solution was not in terms of radicals: that was already proven impossible by the [Abel–Ruffini theorem](/wiki/Abel%E2%80%93Ruffini_theorem "Abel–Ruffini theorem")).

In [algebraic number theory](/wiki/Algebraic_number_theory "Algebraic number theory") Kronecker introduced the [theory of divisors](/wiki/Divisor_%28algebraic_geometry%29 "Divisor (algebraic geometry)") as an alternative to [Dedekind](/wiki/Richard_Dedekind "Richard Dedekind")'s theory of [ideals](/wiki/Ideal_%28ring_theory%29 "Ideal (ring theory)"), 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.

Kronecker also contributed to the concept of [continuity](/wiki/Continuity_%28mathematics%29 "Continuity (mathematics)"), reconstructing the form of [irrational numbers](/wiki/Irrational_number "Irrational number") in [real numbers](/wiki/Real_number "Real number"). In [analysis](/wiki/Mathematical_analysis "Mathematical analysis"), Kronecker rejected the formulation of a [continuous, nowhere differentiable function](/wiki/Continuous%2C_nowhere_differentiable_function "Continuous, nowhere differentiable function") by his colleague, [Karl Weierstrass](/wiki/Karl_Weierstrass "Karl Weierstrass").

Also named for Kronecker are the [Kronecker limit formula](/wiki/Kronecker_limit_formula "Kronecker limit formula"), [Kronecker's congruence](/wiki/Kronecker%27s_congruence "Kronecker's congruence"), [Kronecker delta](/wiki/Kronecker_delta "Kronecker delta"), [Kronecker comb](/wiki/Kronecker_delta%23The_Kronecker_comb "Kronecker delta#The Kronecker comb"), [Kronecker symbol](/wiki/Kronecker_symbol "Kronecker symbol"), [Kronecker product](/wiki/Kronecker_product "Kronecker product"), [Kronecker's method](/wiki/Kronecker%27s_method "Kronecker's method") for factorizing polynomials, [Kronecker substitution](/wiki/Kronecker_substitution "Kronecker substitution"), [Kronecker's theorem](/wiki/Kronecker%27s_theorem "Kronecker's theorem") in number theory, [Kronecker's lemma](/wiki/Kronecker%27s_lemma "Kronecker's lemma"), and [Eisenstein–Kronecker numbers](/wiki/Eisenstein%E2%80%93Kronecker_number "Eisenstein–Kronecker number").

### Philosophy of mathematics

Kronecker's [finitism](/wiki/Finitism "Finitism") made him a forerunner of [intuitionism](/wiki/Intuitionism "Intuitionism") in [foundations of mathematics](/wiki/Foundations_of_mathematics "Foundations of mathematics").


What implications do Kronecker's ideas have for the foundations of mathematics?