代数数最小多项式[Algebraic Number Minimal Polynomial][2005-9-17] iamet 发表在 ∑〖数学〗
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 using MinimalPolynomial[zeta, var] in the Mathematica add-on package NumberTheory`AlgebraicNumberFields` (which can be loaded with the command <<NumberTheory`) .
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 .
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 using MinimalPolynomial[zeta, var] in the Mathematica add-on package NumberTheory`AlgebraicNumberFields` (which can be loaded with the command <<NumberTheory`) . 
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 
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. 
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 
.