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 pK,α 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 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 pK,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 = G, and containing α, any element g 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 is K-isomorphic to K 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, as the set of elements g for g in Aut. The number of repeats in that list of each element is the separable degree 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 on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.