In 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 , 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, since the algebraic conjugates over

\R

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 fo ...

are the number itself and its ''complex conjugate''.

Example

The cube roots of the number one are: :

-\frac-\fraci \end

The latter two roots are conjugate elements in with minimal polynomial :

\left(x+\frac\right)^2+\frac=x^2+x+1.

Properties

If ''K'' is given inside an algebraically closed field ''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 ''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

s have absolute value at most 1, then ''α'' is a

root 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 ...

. 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)