# 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 **Q**[*i*√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.