A
Kummer extension is a field extension
L/
K, where for some given integer
n > 1 we have •
K contains
n distinct
nth
roots of unity (i.e., roots of
Xn − 1) •
L/
K has
abelian Galois group of
exponent n. 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 L= K(\sqrt{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. 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
X3 −
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
nth roots of unity, which implies that the characteristic of
K doesn't divide
n, then adjoining to
K the
nth 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 \sqrt[n]{a}.
Kummer theory provides converse statements. When
K contains
n distinct
nth 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 :K^{\times}/(K^{\times})^n, that is, elements of
K×
modulo nth powers. The correspondence can be described explicitly as follows. Given a subgroup :\Delta \subseteq K^{\times}/(K^{\times})^n, the corresponding extension is given by :K \left (\Delta^{\frac{1}{n}} \right), where :\Delta^{\frac{1}{n}} = \left \{ \sqrt[n]{a}:a\in K^{\times}, a \cdot \left (K^{\times} \right )^n \in \Delta \right \}. In fact it suffices to adjoin
nth 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 :\Delta = \left (K^\times \cap (L^\times)^n \right )/(K^{\times})^n. In this case there is an isomorphism :\Delta \cong \operatorname{Hom}_{\text{c}}(\operatorname{Gal}(L/K), \mu_n) given by :a \mapsto \left(\sigma \mapsto \frac{\sigma(\alpha)}{\alpha}\right), where α is any
nth root of
a in
L. Here \mu_n denotes the multiplicative group of
nth roots of unity (which belong to
K) and \operatorname{Hom}_{\text{c}}(\operatorname{Gal}(L/K), \mu_n) is the group of continuous homomorphisms from \operatorname{Gal}(L/K) equipped with
Krull topology to \mu_n with discrete topology (with group operation given by pointwise multiplication). This group (with discrete topology) can also be viewed as
Pontryagin dual of \operatorname{Gal}(L/K), assuming we regard \mu_n as a subgroup of
circle group. If the extension
L/
K is finite, then \operatorname{Gal}(L/K) is a finite
discrete group and we have :\Delta \cong \operatorname{Hom}(\operatorname{Gal}(L/K), \mu_n) \cong \operatorname{Gal}(L/K), however the last isomorphism isn't
natural.
Recovering from a primitive element For p prime, let K be a field containing \zeta_p and K(\beta)/K a degree p Galois extension. Note the Galois group is cyclic, generated by \sigma. Let :\alpha= \sum_{l=0}^{p-1} \zeta_p^{l} \sigma^l(\beta) \in K(\beta) If we choose \beta appropriately (for example, if \{\sigma^i(\beta)\}_{i=0}^{p-1} is a
Normal basis), then \alpha\neq 0 and :\zeta_p \sigma(\alpha) = \sum_{l=0}^{p-1} \zeta_p^{l+1} \sigma^{l+1}(\beta) = \alpha. Hence \alpha\ne \sigma(\alpha), so K(\alpha) = K(\beta) and :\alpha^p = \pm \prod_{l=0}^{p-1} \zeta_p^{-l} \alpha = \pm \prod_{l=0}^{p-1} \sigma^l(\alpha) = \pm N_{K(\beta)/K}(\alpha) \in K, where the \pm sign is + if p is odd and - if p=2. When L/K is an abelian extension of degree n= \prod_{j=1}^m p_j square-free such that \zeta_n \in K, apply the same argument to the subfields K(\beta_j)/K Galois of degree p_j to obtain :L = K \left (a_1^{1/p_1},\ldots,a_m^{1/p_m} \right ) = K \left (A^{1/p_1},\ldots,A^{1/p_m} \right )= K \left (A^{1/n} \right ) where :A = \prod_{j=1}^m a_j^{n/p_j} \in K. == Kummer map ==