Algebraic conjugate

An algebraic number is any real number $$\xi$$ that's the root of some polynomial with rational coefficients. That is to say, there exist rational $$a_0,\ldots,a_n$$ such that
 * $$a_0+a_1\xi+a_2\xi^2+\ldots+a_n\xi^n=0.$$

Every algebraic number has an associated minimal polynomial, which is the monic polynomial (its leading coefficient is 1) of least degree of which it is a root. Two algebraic numbers are said to be (algebraically) conjugate if they have the same minimal polynomial. Being conjugate is an equivalence relation.

Algebraic conjugates are related to conjugate polytopes, since the coordinates of conjugate polytopes are always algebraic conjugates of one another. The reason is that automorphisms of fields of real numbers must necessarily send numbers to algebraic conjugates, as they must satisfy the same algebraic equations involving rational numbers.