Kummer theory

From Example Problems
Jump to navigation Jump to search

In mathematics, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field.

The theory was originally developed by Ernst Kummer around the 1840s in his pioneering work on Fermat's Last Theorem.

Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is something much more serious.

Kummer extensions

A Kummer extension of fields is a field extension

where for some given integer n > 1 we have [L:K] = n and

  • L is generated over K by a root of a polynomial Xna with a in K, and
  • K contains n distinct roots of Xn − 1.

For example, when n = 2, the second condition is always true if K has characteristic ≠ 2. The Kummer extensions in this case are all quadratic extensions

where a in K is a non-square element. By the usual solution of quadratic equations, any extension of degree 2 of K has this form. When K has characteristic 2, there are no such Kummer extensions.

Taking n = 3, there are no degree three Kummer extensions of the rational number field Q, since for three cube roots of 1 complex numbers are required. If one takes L to be the splitting field of Xna over Q, where a is not a cube in the rational numbers, then L contains a subfield K with three cube roots of 1; that is because if α and β are roots of the cubic polynomial, we shall have

and the cubic is a separable polynomial. Then L/K is a Kummer extension.

More generally, it is true that when K contains n distinct roots of unity, which implies that the characteristic of K doesn't divide n, then adjoining to K the n-th root of any element a of K creates a Kummer extension (of degree m, for some m dividing n). All such extensions are Galois, with Galois group that is cyclic of order m. In fact it is easy to track the Galois action via the root of unity in front of

Kummer theory

Kummer theory provides converse statements. When K contains n distinct roots of unity, it states that any cyclic extension of K of degree n is formed by extraction of an n-th root. Further, if K× denotes the multiplicative group of non-zero elements of K, cyclic extensions of K of degree n correspond bijectively with cyclic subgroups of

that is, elements of K× modulo n-th powers. The correspondence can be described explicitly as follows. Given a cyclic subgroup

the corresponding extension is given by

that is, by adjoining nth roots of elements of Δ to K. Conversely, if L is a Kummer extension of K, then Δ is recovered by the rule

In this case there is an isomorphism

given by

where α is any nth root of a in L.

Generalizations

There exists a slight generalization of Kummer theory which deals with abelian extensions with Galois group of exponent n, and an analogous statement is true in this context. Namely, one can prove that such extensions are in one-to-one correspondence with subgroups of

that are themselves of exponent n.

The theory of cyclic extensions when the characteristic of K does divide n is called Artin-Schreier theory.

See also