The minimal polynomial of an algebraic number 
is the unique irreducible
polynomial 
of smallest polynomial
degree that is monic and has rational coefficients such that 
.
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 ![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
nth 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 polynomial
degree among all polynomials 
with this property. Its polynomial
degree is equal to the extension field degree
of the extension field 
over 
.
