Kummer theory
Encyclopedia
In abstract algebra
and number theory
, Kummer theory provides a description of certain types of field extension
s involving the adjunction
of nth 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 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.
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(√a) 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 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
that is, elements of K× modulo
nth powers. The correspondence can be described explicitly as follows. Given a cyclic subgroup
the corresponding extension is given by
that is, by adjoining nth roots of elements of Δ to K. 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 nth root of a in L.
s 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
If the ground field K does not contain the nth roots of unity, one sometimes still uses the phrase Kummer theory to refer to the isomorphism
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Kummer theory provides a description of certain types of field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
s involving the adjunction
Adjunction (field theory)
In abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.- Definition :...
of nth roots of elements of the base field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. The theory was originally developed by Ernst Eduard Kummer
Ernst Kummer
Ernst Eduard Kummer 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 high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
around the 1840s in his pioneering work on Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. 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
Artin-Schreier theory
In mathematics, Artin–Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p...
.
Kummer theory is basic, for example, in class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
and in general in understanding abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
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 nth roots of unityRoot of unityIn mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
(i.e., roots of Xn-1) - L/K has abelianAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
Galois groupGalois groupIn mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
of exponent n.
For example, when n = 2, the first condition is always true if K has characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
≠ 2. The Kummer extensions in this case include quadratic extensions L = K(√a) where a in K is a non-square element. By the usual solution of quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
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
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
field Q, since for three cube roots of 1 complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s 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
Separable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
. 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
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
of the polynomial Xn − a, the Kummer extension is necessarily Galois
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
, with Galois group that is cyclic
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order m. It is easy to track the Galois action via the root of unity in front of
Kummer theory
Kummer theory provides converse statements. When K contains n distinct nth roots of unity, it states that any cyclic extension of K of degree n is formed by extraction of an nth 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 ofthat is, elements of K× modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
nth powers. The correspondence can be described explicitly as follows. Given a cyclic subgroup
the corresponding extension is given by
that is, by adjoining nth roots of elements of Δ to K. 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 nth root of a in L.
Generalizations
There exists a slight generalization of Kummer theory which deals with abelian extensionAbelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
s 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
If the ground field K does not contain the nth roots of unity, one sometimes still uses the phrase Kummer theory to refer to the isomorphism