Kummer theory

In mathematics, Kummer theory provides a description of certain types of field extensions involving the adjunction of "n"th 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 "L"/"K", where for some given integer "n" > 1 we have ["L":"K"] = "n" and moreover we have:

*"L" is generated over "K" by a root of a polynomial "X"^"n" − "a" with "a" in "K".
*"K" contains "n" distinct roots of "X"^"n" − 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 "L" = "K"(√a) 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 "X"^"n" − "a" 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 (α/β)³ =1 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 "n"th 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"). As the splitting field of the polynomial "X"^"n" − "a", the Kummer extension is necessarily Galois, with Galois group that is cyclic of order "m". It is easy to track the Galois action via the root of unity in front of $sqrt\; [n]\; \{a\}.$

Kummer theory

Kummer theory provides converse statements. When "K" contains "n" distinct "n"th 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

:$K^\{\; imes\}/(K^\{\; imes\})^n,,!$

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

:$Delta\; subseteq\; K^\{\; imes\}/(K^\{\; imes\})^n,\; ,!$

the corresponding extension is given by

:$K(Delta^\{1/n\}),,!$

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

:$Delta\; =\; K\; cap\; L^n.,!$

In this case there is an isomorphism

:$Delta\; cong\; operatorname\{Hom\}(operatorname\{Gal\}(L/K),\; mu\_n)$

given by

:$a\; mapsto\; (sigma\; mapsto\; frac\{sigma(alpha)\}\{alpha\}),$

where α is any "n"th 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

:$K^\{\; imes\}/(K^\{\; imes\})^n\; ,!$

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

* Quadratic field

References

* Bryan Birch, "Cyclotomic fields and Kummer extensions", in J.W.S. Cassels and A. Frohlich (edd), "Algebraic number theory", Academic Press, 1973. Chap.III, pp.45-93.