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 Wolfram Language
package AlgebraicNumberFields` .
For example, the minimal polynomial of 
is 
. In general, the minimal polynomial of ![RadicalBox[p, n]](/images/equations/AlgebraicNumberMinimalPolynomial/Inline6.svg)
, 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 
.
