In abstract algebra, an abelian extension is a Galois extension for which the associated Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. 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. In general a cyclotomic extension formed by adjoining roots of unity is abelian. The cyclotomic fields are examples. 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. 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.
