 Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable.
Any finite extension of a finite field is a cyclic extension. 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.
There are two slightly different concepts of cyclotomic extensions: these can mean either extensions formed by adjoining roots of unity, or subextensions of such extensions. The cyclotomic fields are examples. Any cyclotomic extension (for either definition) is abelian.
If a field K contains a primitive nth root of unity and the nth root of an element of K is adjoined, the resulting socalled 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 nth roots of elements operate both on the nth roots and on the roots of unity, giving a nonabelian Galois group as semidirect 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.
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.
References
 Kuz'min, L.V. (2001), "cyclotomic extension", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/C/c027560.htm
Categories: Field extensions
 Algebraic number theory
 Class field theory
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