Minimum polynomial of an algebraic number

### 7.5.2 Minimum polynomial of an algebraic number

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 11.1.1).

  pmin(α,x) returns the minimal polynomial for α as a symbolic expression with the variable x.

#### Examples

pmin(sqrt(2)+sqrt(3))

▯ 1,0,−10,0,1▯ |  |  |  |  |  |  |  |  |  |

pmin(sqrt(2)+sqrt(3),x)

x4−10 x2+1 |  |  |  |  |  |  |  |  |  |

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.

pmin(sqrt(2)+i*sqrt(3))

▯ 1,0,2,0,25▯ |  |  |  |  |  |  |  |  |  |

pmin(sqrt(2)+i*sqrt(3),z)

z4+2 z2+25 |  |  |  |  |  |  |  |  |  |

pmin(sqrt(2)+2*i)

▯ 1,0,4,0,36 ▯ |  |  |  |  |  |  |  |  |  |

pmin(sqrt(2)+2*i,z)

z4+4 z2+36 |  |  |  |  |  |  |  |  |  |