Kummer theory

In the mathematical subfield of field theory the grief theory describes certain field extensions obtained by adjoining -th roots of elements of the body. Originally, the theory of Ernst Eduard Kummer was developed during his employment with the Fermat conjecture in the 1840s.

The main statements of the theory do not depend on the particular body, whose characteristic may only be a divisor of. A fundamental role is played by the Kummer theory in class field theory, generally it is to the understanding of abelian extensions important; she states that cyclical expansions can be obtained by extraction of roots, provided that the body contains enough roots of unity.

Kummer extensions

Definition

Let be a natural number. A Kummer extension is a field extension, for which:

• Contains various -th roots of unity, ie the zeros of the polynomial.
• Has an abelian Galois group of exponent. The latter means that the Galois true for all elements.

Examples

• Is, the initial condition is always satisfied if the characteristic is not 2, the two unit 1 and roots. Kummer extensions in this case are initially quadratic extensions, with a non-square element of is. The solution formula for quadratic equations shows that any extension of degree 2 has this form. Also Kummer extensions are biquadratic ( two by adjoining square roots ) and general multiquatratische ( by adjoining square roots of several ) extensions. Has the characteristic 2, there is no sorrow extensions, since the equation is valid in characteristic 2, ie there are no two different roots of unity.

• For there is no Kummer extensions of the rational numbers, since not all three third roots of unity are rational. Be an arbitrary rational number that is not a cube, and the splitting field of over. Are and zeros of this cubic polynomial, the following applies. Since the cubic polynomial is also separable, it has three distinct zeros. So are the two non-trivial third roots of unity, namely and, in so a lower body possesses, which contains the three roots of unity. Then is a Kummer extension.

• Contains various general -th roots of unity, from which already follows that the characteristic of not divide is, we get by adjoining a - th root of an element of the body a grief extension. Your degree is a divisor of. As a splitting field of the polynomial expansion of the grief is automatically Galois with cyclic Galois group of order.

Kummer theory

The Kummer theory makes statements the reverse direction. If a body is, the different -th roots of unity contains, it says that every cyclic extension of can be obtained by drawing a - th root of degree. If we denote by the multiplicative group of nonzero elements of the body, so are the cyclic extensions of degree, which lie in a fixed chosen algebraic degree, in bijection with the cyclic subgroups of, so the factor group of under - th powers.

The bijection can be specified explicitly: one cyclic subgroup is assigned to the extension that arises from to by adjoining all - th roots of elements.

Conversely assigns you the grief extension to the subgroup.

Assigns this bijection the group and the body extension to each other, so there is an isomorphism of which is given. This is the group of roots of unity and for an arbitrary - th root of.

Generalizations

The above correspondence continues to be a bijection between subgroups and abelian extensions of exponent. This general version was first reported by Ernst Witt.

In characteristic there is an analogous theory for cyclic extensions of degree, the Artin - Schreier theory. A generalization for abelian extensions of exponent is also from Witt. It uses imported in the same work Witt vectors.

Footnotes

Swell

• Jürgen Neukirch: class field theory. Bibliographical Institute, Mannheim 1986.
• L. V. Kuz'min: Kummer extension. In: Michiel Hazewinkel (ed.): Encyclopaedia of Mathematics. Springer -Verlag, Berlin 2002, ISBN 1-4020-0609-8 ( online ).

• Body theory
• Algebraic Number Theory