Kronecker?Weber theorem

In algebraic number theory, the Kronecker?Weber theorem states that every finite abelian extension of the field of rational numbers Bbb{Q}, or in other words every algebraic number field whose Galois group over Bbb{Q} is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root of unity to the rational numbers.

Related Topics:
Algebraic number theory - Abelian extension - Rational number - Algebraic number field - Galois group - Cyclotomic field - Root of unity

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Kronecker provided most of the proof in 1853, with Weber in 1886 and Hilbert in 1896 filling in the gaps. It can be proven by a straightforward algebraic construction, though it is also an easy consequence of class field theory and can be proven by putting together local data over the p-adic fields for each prime p.

Related Topics:
Kronecker - Weber - Hilbert - Class field theory - P-adic field

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For a given abelian extension K of Q there is in fact a minimal cyclotomic field that contains it. The theorem allows one to define the conductor f of K, as the smallest integer n such that K lies inside the field generated by the n-th roots of unity. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact broadly generalised in class field theory.

Related Topics:
Conductor - Quadratic field - Discriminant

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