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# Conjugate Elements

Two elements , of a field , which is an extension field of a field , are called conjugate (over ) if they are both algebraic over and have the same minimal polynomial.

Two complex conjugates and () are also conjugate in this more abstract meaning, since they are the roots of the following monic polynomial

 (1)

with real coefficients, which is irreducible since its discriminant is negative, and hence is their common minimal polynomial over the field of real numbers.

All primitive th roots of unity are conjugate over since they have the cyclotomic polynomial as their common minimal polynomial. So, for, instance, the primitive fifth roots of unity

 (2) (3) (4) (5)

are all conjugate over . This shows that elements (such as and ) which are not conjugate over a larger field () may be conjugate over a smaller field.

The number of conjugates of an algebraic element over is less than or equal to the degree of its minimal polynomial over , and equality holds iff has no multiple roots in its splitting field (which is always the case for or ). For example, the minimal polynomial of over is

 (6)

which has 4 simple roots in its splitting field :

 (7)

These are the conjugates of over .

This conjugacy relation is an equivalence relation on the set of algebraic elements in a given extension of the field . Every element of the Galois group of the field extension maps each conjugacy class to itself, permuting its elements.