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abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, Kummer theory provides a description of certain types of
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s involving the adjunction of ''n''th roots of elements of the base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. The theory was originally developed by
Ernst Eduard Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...
around the 1840s in his pioneering work on
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
. 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 In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for ex ...
. Kummer theory is basic, for example, in
class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
and in general in understanding
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
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
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
(i.e., roots of ''X''''n'' − 1) *''L''/''K'' has abelian
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
''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) where ''a'' in ''K'' is a non-square element. By the usual solution of
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
field Q, since for three cube roots of 1
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s 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 In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely ...
. 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 In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
of the polynomial ''Xn'' − ''a'', the Kummer extension is necessarily Galois, with Galois group that is
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
of order ''m''. It is easy to track the Galois action via the root of unity in front of \sqrt Kummer theory provides converse statements. When ''K'' contains ''n'' distinct ''n''th roots of unity, it states that any
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
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^/(K^)^n, that is, elements of ''K''× modulo ''n''th powers. The correspondence can be described explicitly as follows. Given a subgroup :\Delta \subseteq K^/(K^)^n, the corresponding extension is given by :K \left (\Delta^ \right), where :\Delta^ = \left \. 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 :\Delta = \left (K^\times \cap (L^\times)^n \right )/(K^)^n. In this case there is an isomorphism :\Delta \cong \operatorname_(\operatorname(L/K), \mu_n) given by :a \mapsto \left(\sigma \mapsto \frac\right), where α is any ''n''th root of ''a'' in ''L''. Here \mu_n denotes the multiplicative group of ''n''th roots of unity (which belong to ''K'') and \operatorname_(\operatorname(L/K), \mu_n) is the group of continuous homomorphisms from \operatorname(L/K) equipped with
Krull topology In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
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 In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
of \operatorname(L/K), assuming we regard \mu_n as a subgroup of
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ...
. If the extension ''L''/''K'' is finite, then \operatorname(L/K) is a finite discrete group and we have :\Delta \cong \operatorname(\operatorname(L/K), \mu_n) \cong \operatorname(L/K), however the last isomorphism isn't
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
.


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_^ \zeta_p^ \sigma^l(\beta) \in K(\beta) Then :\zeta_p \sigma(\alpha) = \sum_^ \zeta_p^ \sigma^(\beta) = \alpha. Since \alpha\ne \sigma(\alpha), K(\alpha) = K(\beta) and :\alpha^p = \pm \prod_^ \zeta_p^ \alpha = \pm \prod_^ \sigma^l(\alpha) = \pm N_(\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_^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^,\ldots,a_m^ \right ) = K \left (A^,\ldots,A^ \right )= K \left (A^ \right ) where :A = \prod_^m a_j^ \in K.


The Kummer Map

One of the main tools in Kummer theory is the Kummer map. Let m be a positive integer and let K be a field, not necessarily containing the mth roots of unity. Letting \overline denote the algebraic closure of K, there is a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
0\xrightarrow \overline^ \xrightarrow \overline^ \xrightarrow \overline^\xrightarrow 0 Choosing an extension L/K and taking \mathrm(\overline/L)-cohomology one obtains the sequence 0\xrightarrow L^/(L^)^ \xrightarrow H^1\left(L, \overline^ right) \xrightarrow H^1\left(L,\overline^\right) xrightarrow0 By
Hilbert's Theorem 90 In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is an extension of ...
H^1\left(L,\overline^\right)=0, and hence we get an isomorphism \delta: L^/\left(L^\right)^m\xrightarrowH^1\left(L,\overline^ right). This is the Kummer map. A version of this map also exists when all m are considered simultaneously. Namely, since L^/(L^)^m=L^\otimes m^ \mathbb/\mathbb, taking the direct limit over m yields an isomorphism \delta: L^ \otimes \mathbb/\mathbb \xrightarrow H^1\left(L, \overline_\right) , 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 E/K be an elliptic curve. There is a short exact sequence 0\xrightarrow E xrightarrow E \xrightarrow E \xrightarrow 0, where the multiplication by m map is surjective since E is divisible. Choosing an algebraic extension L/K and taking cohomology, we obtain the Kummer sequence for E: 0\xrightarrow E(L)/mE(L)\xrightarrow H^1(L, E \xrightarrow H^1(L,E) xrightarrow0. The computation of the weak Mordell-Weil group E(L)/mE(L) is a key part of the proof of the Mordell-Weil theorem. The failure of H^1(L,E) 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 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 In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for ex ...
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 In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
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''.


See also

*
Quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...


References

* {{Springer, title = Kummer extension, id = k/k055960 *
Bryan Birch Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture. Biography Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. ...
, "Cyclotomic fields and Kummer extensions", in
J.W.S. Cassels John William Scott "Ian" Cassels, FRS (11 July 1922 – 27 July 2015) was a British mathematician. Biography Cassels was educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh. He went on to study a ...
and
A. Frohlich A is the first letter of the Latin and English alphabet. A may also refer to: Science and technology Quantities and units * ''a'', a measure for the attraction between particles in the Van der Waals equation * A value, ''A'' value, a mea ...
(edd), ''Algebraic number theory'',
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference ...
, 1973. Chap.III, pp. 85–93. Field (mathematics) Algebraic number theory