Conjugate element (field theory)

Template:Refimprove In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element Template:Math, over a field extension Template:Math, are the roots of the minimal polynomial Template:Math of Template:Math over Template:Math. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally Template:Math itself is included in the set of conjugates of Template:Math.

Equivalently (if Template:Math is normal), the conjugates of Template:Math are the images of Template:Math under the field automorphisms of Template:Mvar that leave fixed the elements of Template:Mvar. 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 $\mathbb{ℝ}$ of a complex number are the number itself and its complex conjugate.

Example

The cube roots of unity are:

$\sqrt[3]{1}=\{\begin{array}{l}1\\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i\end{array}$

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

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

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 FTemplate:' that maps polynomial p to pTemplate:' can be extended to an isomorphism of the splitting fields of p over F and pTemplate:' over FTemplate:', 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.