charpoly -- Characteristic and minimal polynomials over the prime field

Synopsis

Usage:

charpoly a

minpoly a
Inputs:
- a, an element of a field
Optional inputs:
- Variable => a symbol, default value x, the variable to use for the polynomial
Outputs:
- the characteristic / minimal polynomial of a over its prime field

Obtain the characteristic / minimal polynomial of an element over its prime field.

i1 : QQ[x]; F = splittingField((x^2+1)*(x^2-2));

i3 : minpoly a

      4     2
o3 = x  - 2x  + 9

o3 : QQ[x]

i4 : charpoly(a^2+1, Variable=>y)

      4     3      2
o4 = y  - 8y  + 40y  - 96y + 144

o4 : QQ[y]

i5 : minpoly(a^2+1, Variable=>y)

      2
o5 = y  - 4y + 12

o5 : QQ[y]

i6 : GF 81; minpoly(a+1)

      4    3
o7 = x  + x  - x + 1

     ZZ
o7 : --[x]
      3

The method minpoly can also be used on a field to recover the polynomial used in its definition.

i8 : minpoly F

      4     2
o8 = a  - 2a  + 9

o8 : QQ[a]