mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in particular field theory, the conjugate elements or algebraic conjugates of an

algebraic element In mathematics, if is an associative algebra over , then an element of is an algebraic element over , or just algebraic over , if there exists some non-zero polynomial g(x) \in K /math> with coefficients in such that . Elements of that are no ...

, over a

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

, are the

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

s 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 (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

. 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 numbers, then the complex conjugate of a + bi is a - ...

, 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 for ...

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

Example

The cube roots of the number

one 1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...

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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

''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 polyn ...

over ''K'' of ''p''_''K'',''α'', containing ''α''. If ''L'' is any

normal extension In abstract algebra, a normal extension is an Algebraic extension, algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' that has a zero of a function, root in ''L'' splits into linear factors over ''L''. This is ...

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 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 that 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 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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

at most 1, then ''α'' is a

root of unity In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...

. 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)