Every abelian extension of the rational numbers is contained in some cyclotomic field, a field generated by roots of unity. It is the simplest case of the broader Hilbert twelfth problem on explicit class field theory.
Facts
StatementEvery finite abelian extension of the rational numbers is a subfield of some cyclotomic field, the field generated by adjoining a root of unity to the rationals. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
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In Branch
Sources
1. Kronecker-Weber Theorem (Wikipedia)
Wikimedia FoundationField-theoretic formulation section
Precisely, the Kronecker-Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field.
History section
The theorem was first stated by Kronecker (1853) though his argument was not complete for extensions of degree a power of 2. Weber (1886) published a proof, but this had some gaps and errors that were pointed out and corrected by Neumann (1981). The first complete proof was given by Hilbert (1896).
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