| Abelian Extension |
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| CATEGORIES ABOUT ABELIAN EXTENSION | |
| field theory | |
| algebraic number theory | |
| class field theory | |
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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 Field s, Function Field s of Algebraic Curve s over finite fields, and Local Field s. In general a cyclotomic extension formed by adjoining Roots Of Unity is abelian. The Cyclotomic Field s 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 Number s, 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 which relates directly to the first Homology Group . |
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