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Kummer theory (Theorem)

The following theorem is usually referred to as Kummer theory.

Theorem 1 (Kummer Theory)   Let $ n$ be a positive integer and let $ K$ be a field of characteristic not dividing $ n$ which contains the $ n$-th roots of unity. Then:
  1. The extension $ K(\sqrt[n]{a})$ for $ a\in K$ is a cyclic extension over $ K$ of degree dividing $ n$.
  2. Any cyclic extension of degree $ n$ over $ K$ is of the form $ K(\sqrt[n]{a})$ for some $ a\in K$.
Definition 1   Let $ n$ be a positive integer and let $ K$ be a field of characteristic not dividing $ n$ which contains the $ n$-th roots of unity. An extension of $ K$ of the form:
$\displaystyle K(\sqrt[n]{a_1},\sqrt[n]{a_2},\ldots,\sqrt[n]{a_k})$
with $ a_i \in K^\times$ is called a Kummer extension of $ K$. Notice that the Galois group of the extension is of exponent $ n$.



"Kummer theory" is owned by alozano.
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See Also: abelian extension, cyclic extension, exponent

Also defines:  Kummer extension
Keywords:  Kummer, abelian extension
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Cross-references: Galois group, degree, cyclic extension, extension, roots of unity, contains, characteristic, field, integer, positive, theory
There are 4 references to this entry.

This is version 2 of Kummer theory, born on 2005-02-22, modified 2005-02-22.
Object id is 6793, canonical name is KummerTheory.
Accessed 3430 times total.

Classification:
AMS MSC: 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)
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