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

## Synopsis

• Usage:
minimalPolynomial(f)
minimalPolynomial(g,I)
• Inputs:
• f, , an element of an Artinian ring
• g, , a polynomial
• I, an ideal, a zero-dimensional ideal
• Optional inputs:
• Strategy => ..., default value 0, set method for computing the minimal polynomial
• Outputs:
• , the desired minimal polynomial. See description

## Description

This computes the minimal polynomial of a ring element f in the Artinian ring ring f, or the minimal polynomial of a polynomial g in the Artinian ring (ring g)/I. When f is a variable in ring f, this is the eliminant with respect to that variable.

 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]

We provide two examples to compute minimal polynomials given by Strategy => 0 (computes the kernel of $k[T]\to$ ring f by sending $T$ to f) and Strategy => 1 (a minimal linear combination of powers of the input).

 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]

## Ways to use minimalPolynomial :

• "minimalPolynomial(RingElement)"
• "minimalPolynomial(RingElement,Ideal)"

## For the programmer

The object minimalPolynomial is .