minimalPolynomial -- the minimal polynomial of an element of an Artinian ring

• Usage:

  minimalPolynomial(f)

  minimalPolynomial(g,I)

• Inputs:
  • f, a ring element, an element of an Artinian ring
  • g, a ring element, a polynomial
  • I, an ideal, a zero-dimensional ideal in a polynomial ring
• Optional inputs:
  • Strategy => ..., default value 0, set method for computing the minimal polynomial
  • Variable => ..., default value "Z", allows user to change the variable of the resulting polynomial
• Outputs:
  • a ring element, the desired minimal polynomial. See description

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(y^2 - x^2 - 1,x - y^2 + 4*y - 2)

               2    2         2
o2 = ideal (- x  + y  - 1, - y  + x + 4y - 2)

o2 : Ideal of R

i3 : minimalPolynomial(y,I)

      4     3      2
o3 = Z  - 8Z  + 19Z  - 16Z + 5

o3 : QQ[Z]

i4 : S = R/I

o4 = S

o4 : QuotientRing

i5 : minimalPolynomial(y)

      4     3      2
o5 = Z  - 8Z  + 19Z  - 16Z + 5

o5 : QQ[Z]

i6 : minimalPolynomial(x,Strategy => 0)

      4     3     2
o6 = Z  - 2Z  - 9Z  - 6Z - 7

o6 : QQ[Z]

i7 : minimalPolynomial(x,Strategy => 1)

      4     3     2
o7 = Z  - 2Z  - 9Z  - 6Z - 7

o7 : QQ[Z]