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In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker . Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later half of the 20th century. In ...
It uses the theorem that a bounded increasing sequence of real numbers has a limit, which can be proved by using Cantor's or Richard Dedekind's construction of the irrational numbers. Because Leopold Kronecker did not accept these constructions, Cantor was motivated to develop a new proof. [1]
The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in 1870, [citation needed] using a group-theoretic proof, [4] though without stating it in group-theoretic terms; [5] a modern presentation of Kronecker's proof is given in (Stillwell 2012), 5.2.2 Kronecker's Theorem, 176–177.
Rouché–Capelli theorem in English speaking countries, Italy and Brazil; Kronecker–Capelli theorem in Austria, Poland, Ukraine, Croatia, Romania, Serbia and Russia; Rouché–Fontené theorem in France; Rouché–Frobenius theorem in Spain and many countries in Latin America; Frobenius theorem in the Czech Republic and in Slovakia.
Kronecker's theorem on diophantine approximation; Almost periodic function; Bohr compactification; Wiener's tauberian theorem; Representation theory
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field.It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity that generate a whole family of further number fields, analogously to the cyclotomic fields and their subfields.
Krener's theorem (control theory) Kronecker's theorem (Diophantine approximation) Kronecker–Weber theorem (number theory) Krull's principal ideal theorem (commutative algebra) Krull–Schmidt theorem (group theory) Kruskal's tree theorem (order theory) Kruskal–Katona theorem (combinatorics) Krylov–Bogolyubov theorem (dynamical systems)
12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality 13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments. 14. Proof of the finiteness of certain complete systems of functions. 15. Rigorous foundation of Schubert's enumerative calculus. 16.