# Conjugate element (field theory)

In mathematics, in particular field theory, the **conjugate elements** of an algebraic element α, over a field *K*, are the (other) roots of the minimal polynomial

*P*_{K,α}(*t*)

of α over *K*. If *K* is given inside an algebraically closed field *C*, then the conjugates can be taken inside *C*. Usually one includes α itself in the set of conjugates.

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. Assuming only that *P* is a separable polynomial, this means that we can take *L* to be a Galois extension.

Given then a Galois extension *L* of *K*, with Galois group *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 of α is of this form: in other words, *G* acts transitively on the conjugates. This follows because it is true for the Galois group of the splitting field, and *G* maps surjectively to that group by basic properties of the Galois correspondence.

In summary, assuming separability, the conjugate elements of α are found, in any finite Galois extension *L* of *K* that contains *K*(α), as the set of elements *g*(α) for *g* in *Gal*(*L*/*K*). The number of repeats in that list of each element is the degree [*L*:*K(α)*] which is also [*L*:*K*]/*d* where *d* is the degree of *P*.

A theorem of Kronecker states that if α is an algebraic integer such that α and all of its conjugates in the complex numbers have absolute value 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. hu:Ciklikus konjugált