Algebraic Number Minimal Polynomial

The minimal polynomial of an algebraic number zeta is the unique irreducible monic polynomial of smallest degree p(x) with rational coefficients such that p(zeta)=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 RadicalBox[p, 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 Phi_n(x). For example, Phi_3(x)=x^2+x+1 is the minimal polynomial of

In general, two algebraic numbers that are complex conjugates have the same minimal polynomial.

Considering the extension field Q(zeta) as a finite-dimensional vector space over the field of the rational numbers, then multiplication by zeta induces a linear transformation T_zeta on Q(zeta). The matrix minimal polynomial of T_zeta, as a linear transformation, is the same as the minimal polynomial of zeta, as an algebraic number.

A minimal polynomial divides any other polynomial with rational coefficients f(x) such that f(alpha)=0. It follows that it has minimal degree among all polynomials f with this property. Its degree is equal to the degree of the extension field Q(alpha) over Q.

See also

Algebraic Number, Conjugate Elements, Eisenstein's Irreducibility Criterion, Extension Field Minimal Polynomial, Matrix Minimal Polynomial, Splitting Field

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Margherita Barile

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References

Jacobson, N. Algebra. New York: W. H. Freeman, p. 131, 1985.Stewart, I. and Tall, D. Algebraic Number Theory. New York: Chapman and Hall, 1987.

Referenced on Wolfram|Alpha

Algebraic Number Minimal Polynomial

Cite this as:

Barile, Margherita; Rowland, Todd; and Weisstein, Eric W. "Algebraic Number Minimal Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicNumberMinimalPolynomial.html

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