Conjugate element (field theory)
Encyclopedia
In mathematics
, in particular field theory
, the conjugate elements of an algebraic element
α, over a field
K, are the (other) roots of the minimal polynomial
of α over K.
The latter two roots are conjugate elements in the field K = Q[√−3].
C, then the conjugates can be taken inside C. Usually one includes α itself in the set of conjugates. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a splitting field
over K of pK,α, containing α. If L is any normal extension
of K containing α, then by definition it already contains such a splitting field.
Given then a normal extension L of K, with automorphism group
Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism
g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively
on the conjugates. This follows as K(α) is K-isomorphic to K(β) by irreducibility of the minimal polynomial, and any isomorphism of fields F and F that maps polynomial p to p can be extended to an isomorphism of the splitting fields of p over F and p over F, respectively.
In summary, the conjugate elements of α are found, in any normal extension L of K that contains K(α), as the set of elements g(α) for g in Aut(L/K). The number of repeats in that list of each element is the separable degree [L:K(α)]sep.
A theorem of Kronecker states that if α is an algebraic integer
such that α and all of its conjugates in the complex number
s have absolute value
1, then α is a root of unity
. There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in particular field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, the conjugate elements of an algebraic element
Algebraic element
In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g with coefficients in K such that g=0...
α, over a 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...
K, are the (other) roots of the minimal polynomial
Minimal polynomial (field theory)
In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...
- pK,α(x)
of α over K.
Example
The cube roots of the number one are:The latter two roots are conjugate elements in the field K = Q[√−3].
Properties
If K is given inside an algebraically closed fieldAlgebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
C, then the conjugates can be taken inside C. Usually one includes α itself in the set of conjugates. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a 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:...
over K of pK,α, containing α. If L is any normal extension
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
of K containing α, then by definition it already contains such a splitting field.
Given then a normal extension L of K, with automorphism group
Galois group
In 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...
Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
on the conjugates. This follows as K(α) is K-isomorphic to K(β) by irreducibility of the minimal polynomial, and any isomorphism of fields F and F that maps polynomial p to p can be extended to an isomorphism of the splitting fields of p over F and p over F, respectively.
In summary, the conjugate elements of α are found, in any normal extension L of K that contains K(α), as the set of elements g(α) for g in Aut(L/K). The number of repeats in that list of each element is the separable degree [L:K(α)]sep.
A theorem of Kronecker states that if α is an algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
such that α and all of its conjugates in the 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 have absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1, then α is a root of unity
Root of unity
In 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...
. There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.