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Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over Q that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.
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.
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 ...
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions.
1837 Dirichlet's theorem on arithmetic progressions. 1853 Leopold Kronecker announces the Kronecker–Weber theorem; 1880 Kronecker introduces his Jugendtraum about abelian extensions of imaginary quadratic fields; 1886 Heinrich Martin Weber proves the Kronecker–Weber theorem (with a slight gap).
The Kronecker–Weber theorem states that every finite abelian extension of Q in C is contained in Q(ζ n) for some n. Equivalently, the union of all the cyclotomic fields Q(ζ n) is the maximal abelian extension Q ab of Q.
The leader of Ontario, Canada’s most populous province, said Monday he will pause all retaliatory measures against the United States after news broke that the threat of tariffs has been put on ...
Resolved. Result: Impossible; Matiyasevich's theorem implies that there is no such algorithm. 1970 11th: Solving quadratic forms with algebraic numerical coefficients. Partially resolved. [17] — 12th: Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field. Partially resolved. [18] — 13th