Editing Kummer theory

{{Short description|Theory in abstract algebra}}
In [[abstract algebra]] and [[number theory]], '''Kummer theory''' provides a description of certain types of [[field extension]]s involving the [[adjunction (field theory)|adjunction]] of ''n''th roots of elements of the base [[field (mathematics)|field]]. The theory was originally developed by [[Ernst Kummer|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 of a field|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 extension]]s; 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

*''K'' contains ''n'' distinct ''n''th [[root of unity|roots of unity]] (i.e., roots of ''X''<sup>''n''</sup> − 1)
*''L''/''K'' has [[abelian group|abelian]] [[Galois group]] of [[exponent (group theory)|exponent]] ''n''.

For example, when ''n'' = 2, the first condition is always true if ''K'' has [[characteristic (algebra)|characteristic]] &ne; 2. The Kummer extensions in this case include '''quadratic extensions''' <math>L= K(\sqrt{a})</math> where ''a'' in ''K'' is a non-square element. By the usual solution of [[quadratic equation]]s, 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 number]]s are required. If one takes ''L'' to be the splitting field of ''X''<sup>3</sup> − ''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 &alpha; and &beta; are roots of the cubic polynomial, we shall have (&alpha;/&beta;)<sup>3</sup> =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<sup>n</sup>'' − ''a'', the Kummer extension is necessarily [[Galois extension|Galois]], with Galois group that is [[cyclic group|cyclic]] of order ''m''. It is easy to track the Galois action via the root of unity in front of <math>\sqrt[n]{a}.</math>

'''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''<sup>×</sup> denotes the multiplicative group of non-zero elements of ''K'', abelian extensions of ''K'' of exponent ''n'' correspond bijectively with subgroups of

:<math>K^{\times}/(K^{\times})^n,</math>

that is, elements of ''K''<sup>×</sup> [[Modular arithmetic|modulo]] ''n''th powers. The correspondence can be described explicitly as follows. Given a subgroup

:<math>\Delta \subseteq K^{\times}/(K^{\times})^n,</math>

the corresponding extension is given by

:<math>K \left (\Delta^{\frac{1}{n}} \right),</math>

where 

:<math>\Delta^{\frac{1}{n}} = \left \{ \sqrt[n]{a}:a\in K^{\times}, a \cdot \left (K^{\times} \right )^n \in \Delta \right \}.</math> 

In fact it suffices to adjoin ''n''th root of one representative of each element of any set of generators of the group &Delta;. Conversely, if ''L'' is a Kummer extension of ''K'', then &Delta; is recovered by the rule

:<math>\Delta = \left (K^\times \cap (L^\times)^n \right )/(K^{\times})^n.</math>

In this case there is an isomorphism

:<math>\Delta \cong \operatorname{Hom}_{\text{c}}(\operatorname{Gal}(L/K), \mu_n)</math>

given by

:<math>a \mapsto \left(\sigma \mapsto \frac{\sigma(\alpha)}{\alpha}\right),</math>

where &alpha; is any ''n''th root of ''a'' in ''L''. Here <math>\mu_n</math> denotes the multiplicative group of ''n''th roots of unity (which belong to ''K'') and <math>\operatorname{Hom}_{\text{c}}(\operatorname{Gal}(L/K), \mu_n)</math> is the group of continuous homomorphisms from <math>\operatorname{Gal}(L/K)</math> equipped with [[Krull topology]] to <math>\mu_n</math> with discrete topology (with group operation given by pointwise multiplication). This group (with discrete topology) can also be viewed as [[Pontryagin dual]] of <math>\operatorname{Gal}(L/K)</math>, assuming we regard <math>\mu_n</math> as a subgroup of [[circle group]]. If the extension ''L''/''K'' is finite, then <math>\operatorname{Gal}(L/K)</math> is a finite discrete group and we have

:<math>\Delta \cong \operatorname{Hom}(\operatorname{Gal}(L/K), \mu_n) \cong \operatorname{Gal}(L/K),</math>

however the last isomorphism isn't [[Natural homomorphism|natural]].

=== Recovering {{math|1= ''a''<sup>1/''n''</sup> }} from a primitive element ===
For <math>p</math> prime, let <math>K</math> be a field containing <math>\zeta_p</math> and <math>K(\beta)/K</math> a degree <math>p</math> Galois extension. Note the Galois group is cyclic, generated by <math>\sigma</math>. Let 

:<math>\alpha= \sum_{l=0}^{p-1} \zeta_p^{l} \sigma^l(\beta) \in K(\beta)</math>

Then 

:<math>\zeta_p \sigma(\alpha) = \sum_{l=0}^{p-1} \zeta_p^{l+1} \sigma^{l+1}(\beta) = \alpha.</math> 

Since <math>\alpha\ne \sigma(\alpha), K(\alpha) = K(\beta)</math> and 

:<math>\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</math>,
where the <math>\pm</math> sign is <math>+</math> if <math>p</math> is odd and <math>-</math> if <math>p=2</math>.

When <math>L/K</math> is an abelian extension of degree <math>n= \prod_{j=1}^m p_j</math> square-free such that <math>\zeta_n \in K</math>, apply the same argument to the subfields <math>K(\beta_j)/K</math> Galois of degree <math>p_j</math> to obtain 

:<math>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 )</math> 

where 

:<math>A = \prod_{j=1}^m a_j^{n/p_j} \in K</math>.

== The Kummer Map ==
One of the main tools in Kummer theory is the Kummer map. Let <math>m</math> be a positive integer and let <math>K</math> be a field, not necessarily containing the <math>m</math>th roots of unity. Letting <math>\overline{K}</math> denote the algebraic closure of <math>K</math>, there is a [[short exact sequence]]

<math>0\xrightarrow{} \overline{K}^{\times}[m] \xrightarrow{} \overline{K}^{\times} \xrightarrow{z\mapsto z^m} \overline{K}^{\times}\xrightarrow{} 0</math>

Choosing an extension <math>L/K</math> and taking <math>\mathrm{Gal}(\overline{K}/L)</math>-cohomology one obtains the sequence

<math>0\xrightarrow{} L^{\times}/(L^{\times})^{m} \xrightarrow{} H^1\left(L, \overline{K}^{\times}[m]\right) \xrightarrow{} H^1\left(L,\overline{K}^{\times}\right)[m]\xrightarrow{}0</math>

By [[Hilbert's Theorem 90]] <math>H^1\left(L,\overline{K}^{\times}\right)=0</math>, and hence we get an isomorphism <math>\delta: L^{\times}/\left(L^{\times}\right)^m\xrightarrow{\sim}H^1\left(L,\overline{K}^{\times}[m]\right)</math>. This is the Kummer map. A version of this map also exists when all <math>m</math> are considered simultaneously. Namely, since <math>L^{\times}/(L^{\times})^m=L^{\times}\otimes m^{-1} \mathbb{Z}/\mathbb{Z}</math>, taking the direct limit over <math>m</math> yields an isomorphism

<math>\delta: L^{\times} \otimes \mathbb{Q}/\mathbb{Z} \xrightarrow{\sim} H^1\left(L, \overline{K}_{tors}\right) </math>,

where ''tors'' denotes the torsion subgroup of roots of unity.

== For Elliptic Curves ==
Kummer theory is often used in the context of elliptic curves. Let <math>E/K</math> be an elliptic curve. There is a short exact sequence

<math>0\xrightarrow{} E[m]\xrightarrow{} E \xrightarrow{P\mapsto m\cdot P} E \xrightarrow{} 0</math>,

where the multiplication by <math>m</math> map is surjective since <math>E</math> is divisible. Choosing an algebraic extension <math>L/K</math> and taking cohomology, we obtain the Kummer sequence for <math>E</math>:

<math>0\xrightarrow{} E(L)/mE(L)\xrightarrow{} H^1(L, E[m])\xrightarrow{} H^1(L,E)[m]\xrightarrow{}0</math>.

The computation of the weak Mordell-Weil group <math>E(L)/mE(L)</math> is a key part of the proof of the [[Mordell–Weil theorem|Mordell-Weil theorem]].  The failure of <math>H^1(L,E)</math> to vanish adds a key complexity to the theory.

== 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 H<sup>1</sup>(''G'',''A'') is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between ''A''<sup>''G''</sup>/π(''A''<sup>''G''</sup>) 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 ''p<sup>n</sup>''.

== See also ==

* [[Quadratic field]]

==References==

* {{Springer|title = Kummer extension|id = k/k055960}}
* [[Bryan John Birch|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.&nbsp;85–93.

[[Category:Field (mathematics)]]
[[Category:Algebraic number theory]]