Conjugate Elements

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

Two complex conjugates z=a+ib and z^_=a-ib (a,b in R,b!=0) are also conjugate in this more abstract meaning, since they are the roots of the following monic polynomial


with real coefficients, which is irreducible since its discriminant Delta=-4b^2 is negative, and hence is their common minimal polynomial over the field R of real numbers.

All primitive nth roots of unity are conjugate over Q since they have the cyclotomic polynomial Phi_n(x) as their common minimal polynomial. So, for, instance, the primitive fifth roots of unity


are all conjugate over Q. This shows that elements (such as alpha_1 and alpha_2) which are not conjugate over a larger field (R) may be conjugate over a smaller field.

The number of conjugates of an algebraic element over F is less than or equal to the degree of its minimal polynomial p(x) over F, and equality holds iff p(x) has no multiple roots in its splitting field (which is always the case for F=Q or F=R). For example, the minimal polynomial of alpha=i+sqrt(2) over Q is


which has 4 simple roots in its splitting field Q(i,sqrt(2)):

 i+sqrt(2), i-sqrt(2), -i+sqrt(2), -i-sqrt(2).

These are the conjugates of alpha over Q.

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

See also

Algebraic Equation, Algebraic Number, Conjugate Element, Polynomial Roots

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Conjugate Elements." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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