The minimal polynomial of an algebraic number is the unique irreducible monic polynomial of smallest degree with rational coefficients such that 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 is . In general, the minimal polynomial of , where and is a prime number, is , which is irreducible by Eisenstein's irreducibility criterion. The minimal polynomial of every primitive th root of unity is the cyclotomic polynomial . For example, is the minimal polynomial of
In general, two algebraic numbers that are complex conjugates have the same minimal polynomial.
Considering the extension field as a finite-dimensional vector space over the field of the rational numbers, then multiplication by induces a linear transformation on . The matrix minimal polynomial of , as a linear transformation, is the same as the minimal polynomial of , as an algebraic number.
A minimal polynomial divides any other polynomial with rational coefficients such that . It follows that it has minimal degree among all polynomials with this property. Its degree is equal to the degree of the extension field over .
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