Two elements α, β 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 + i b and z^_ = a - i b (a, b element R, b!=0) are also conjugate in this more abstract meaning, since they are the roots of the following monic polynomial p(x) = x^2 - 2a x + a^2 + b^2 with real coefficients, which is irreducible since its discriminant Δ = - 4b^2 is negative, and hence is their common minimal polynomial over the field R of real numbers.

algebraic equation | algebraic number | conjugate element | polynomial roots