Abelian extension

In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension.

Related Topics:
Abstract algebra - Field extension - Galois group - Abelian - Cyclic group

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The development of class field theory has provided detailed information about abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.

Related Topics:
Class field theory - Number field - Function field - Algebraic curve - Finite field - Local field

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In general extensions formed by adjoining any roots of unity are abelian. If a field K already contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting so-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory gives a complete description of the abelian extension case, and the Kronecker-Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.

Related Topics:
Roots of unity - Kummer extension - Separable extension - Direct product - Kummer theory - Kronecker-Weber theorem - Rational number

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There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.

Related Topics:
Fundamental group - Topology - Abelianisation - Homology group

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