The minimal polynomial of an algebraic number ζ is the unique irreducible monic polynomial of smallest degree p(x) with rational coefficients such that p(ζ) = 0 and whose leading coefficient is 1. The minimal polynomial can be computed in the Wolfram Language using MinimalPolynomial[zeta, var]. For example, the minimal polynomial of sqrt(2) is x^2 - 2. In general, the minimal polynomial of p^(1/n), where n>=2 and p is a prime number, is x^n - p, which is irreducible by Eisenstein's irreducibility criterion. The minimal polynomial of every primitive nth root of unity is the cyclotomic polynomial Φ_n(x). For example, Φ_3(x) = x^2 + x + 1 is the minimal polynomial of α = - 1/2 + sqrt(3)/(2i), and α^_ = - 1/2 - sqrt(3)/(2i).

algebraic number | conjugate elements | Eisenstein's irreducibility criterion | extension field minimal polynomial | matrix minimal polynomial | splitting field