Kummer theory

In abstract algebra and number theory, 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 Eduard Kummer around the 1840s in his pioneering work on Fermat's last theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer "n" – and therefore belong to abstract algebra. The theory of cyclic extensions of the field "K" when the characteristic of "K" does divide "n" is called Artin–Schreier theory.
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 is a field extension "L"/"K", where for some given integer "n" > 1 we have
For example, when "n" = 2, the first condition is always true if "K" has characteristic ≠ 2. The Kummer extensions in this case include quadratic extensions formula_1 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. The Kummer extensions in this case also include biquadratic extensions and more general multiquadratic extensions. When "K" has characteristic 2, there are no such Kummer extensions.
Taking "n" = 3, there are no degree 3 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"3 − "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 "Xn" − "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 formula_2
Kummer theory provides converse statements. When "K" contains "n" distinct "n"th roots of unity, it states that any abelian extension of "K" of exponent dividing "n" is formed by extraction of roots of elements of "K". Further, if "K"× denotes the multiplicative group of non-zero elements of "K", abelian extensions of "K" of exponent "n" correspond bijectively with subgroups of
that is, elements of "K"× modulo "n"th powers. The correspondence can be described explicitly as follows. Given a subgroup
the corresponding extension is given by
where 
In fact it suffices to adjoin "n"th root of one representative of each element of any set of generators of the group Δ. 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 "n"th root of "a" in "L". Here formula_10 denotes the multiplicative group of "n"th roots of unity (which belong to "K") and formula_11 is the group of continuous homomorphisms from formula_12 equipped with Krull topology to formula_10 with discrete topology (with group operation given by pointwise multiplication). This group (with discrete topology) can also be viewed as Pontryagin dual of formula_12, assuming we regard formula_10 as a subgroup of circle group. If the extension "L"/"K" is finite, then formula_12 is a finite discrete group and we have
however the last isomorphism isn't natural.
Recovering formula_18 from a primitive element.
For formula_19 prime, let formula_20 be a field containing formula_21 and formula_22 a degree formula_19 Galois extension. Note the Galois group is cyclic, generated by formula_24. Let 
Then 
Since formula_27 and 
When formula_29 is an abelian extension of degree formula_30 square-free such that formula_31, apply the same argument to the subfields formula_32 Galois of degree formula_33 to obtain 
where 
Generalizations.
Suppose that "G" is a profinite group acting on a module "A" with a surjective homomorphism π from the "G"-module "A" to itself. Suppose also that "G" acts trivially on the kernel "C" of π and that the first cohomology group H1("G","A") is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between "A""G"/π("A""G") and Hom("G","C").
Kummer theory is the special case of this when "A" is the multiplicative group of the separable closure of a field "k", "G" is the Galois group, π is the "n"th power map, and "C" the group of "n"th roots of unity. Artin–Schreier theory is the special case when "A" is the additive group of the separable closure of a field "k" of positive characteristic "p", "G" is the Galois group, π is the Frobenius map minus the identity, and "C" the finite field of order "p". Taking "A" to be a ring of truncated Witt vectors gives Witt's generalization of Artin–Schreier theory to extensions of exponent dividing "pn".