6.11.2 Minimum polynomial of an algebraic number: pmin
The minimal polynomial of an algebraic number is the monic polynomial
of smallest degree with integer coefficents which has the algebraic
number as a root.
The pmin command finds the minimum polynomial of an algebraic
number.
-
pmin takes one mandatory argument and one optional
argument:
-
α, an algebraic number.
- Optionally, x, a variable name to use as the
variable in the polynomial.
- pmin(α) returns the minimal polynomial for α,
where the polynomial is given as a list of the coefficients (see
Section 6.27.1).
pmin(α,x) returns the minimal
polynomial for α as a symbolic expression with the variable
x.
Examples.
-
Input:
pmin(sqrt(2) + sqrt(3))
Output:
- Input:
pmin(sqrt(2) + sqrt(3),x)
Output:
Note that (√2 + √3)2 = 5 + 2√6 and so
((√2 + √3)2 − 5)2 = 24, which can be rewritten as
(√2 + √3)4 − 10 (√2 + √3)2 + 1 = 0.
- Input:
pmin(sqrt(2) + i*sqrt(3))
Output:
- Input:
pmin(sqrt(2) + i*sqrt(3),z)
Output:
- Input:
pmin(sqrt(2) + 2*i)
Output:
- Input:
pmin(sqrt(2) + 2*i,z)
Output: