conjugate element (field theory)

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In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element {{math|α}}, over a field extension {{math|L/K}}, are the roots of the minimal polynomial {{math|p_K,α(x)}} of {{math|α}} over {{math|K}}. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally {{math|α}} itself is included in the set of conjugates of {{math|α}}.

Equivalently (if {{math|L/K}} is normal), the conjugates of {{math|α}} are the images of {{math|α}} under the field automorphisms of {{mvar|L}} that leave fixed the elements of {{mvar|K}}. The equivalence of the two definitions is one of the starting points of Galois theory.

The concept generalizes complex conjugation, since the algebraic conjugates over $\R$ of a complex number are the number itself and its complex conjugate.

Example

The cube roots of unity are:

: $\sqrt[3]{1} = \begin{cases}1 \\[3pt] -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\[5pt] -\frac{1}{2}-\frac{\sqrt{3}}{2}i \end{cases}$

The latter two roots are conjugate elements in {{math|Q[i{{sqrt|3}}]}} with minimal polynomial

: $\left(x+\frac{1}{2}\right)^2+\frac{3}{4}=x^2+x+1.$

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 over K of p_K,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 a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute value at most 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.

References

David S. Dummit, Richard M. Foote, Abstract algebra, 3rd ed., Wiley, 2004.

External links

{{MathWorld |title=Conjugate Elements |id=ConjugateElements}}

Category:Field (mathematics)