In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular
field theory, the conjugate elements or algebraic conjugates of an
algebraic element
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic over ...
, over a
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 ...
, are the roots of the
minimal polynomial of over . Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally itself is included in the set of conjugates of .
Equivalently, the conjugates of are the images of under the
field 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 ...
s of that leave fixed the elements of . The equivalence of the two definitions is one of the starting points of
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.
The concept generalizes the
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, since the algebraic conjugates over
of a
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 ...
are the number itself and its ''complex conjugate''.
Example
The cube roots of the number
one are:
:
The latter two roots are conjugate elements in with minimal polynomial
:
Properties
If ''K'' is given inside an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
''C'', then the conjugates can be taken inside ''C''. 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
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 ...
over ''K'' of ''p''
''K'',''α'', containing ''α''. If ''L'' is any
normal extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic e ...
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
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
Aut(''L''/''K'') = ''G'', and containing ''α'', any element ''g''(''α'') for ''g'' in ''G'' will be a conjugate of ''α'', since the
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 ...
''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'')sub>sep.
A theorem of
Kronecker states that if ''α'' is a nonzero
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
such that ''α'' and all of its conjugates in the
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 have
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
at most 1, then ''α'' is a
root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
. 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.
References
*David S. Dummit, Richard M. Foote, ''Abstract algebra'', 3rd ed., Wiley, 2004.
External links
*
{{DEFAULTSORT:Conjugate Element (Field Theory)
Field (mathematics)