# Conjugate element (field theory)

In mathematics, in particular field theory, the **conjugate elements** of an algebraic element α, over a field extension *L/K*, are the roots of the minimal polynomial *p*_{K,α}(*x*) of α over *K*.
Conjugate elements are also called *Galois conjugates*, or simply *conjugates*. Normally α itself is included in the set of conjugates of α.

## Example

The cube roots of the number one are:

The latter two roots are conjugate elements in *L*/*K* = **Q**[√3, *i*]/**Q**[√3] with minimal polynomial

## 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.