# MinimalPolynomial

MinimalPolynomial[s,x]

gives the minimal polynomial in x for which the algebraic number s is a root.

MinimalPolynomial[u,x]

gives the minimal polynomial of the finite field element u over .

MinimalPolynomial[u,x,k]

gives the minimal polynomial of u over the -element subfield of the ambient field of u.

MinimalPolynomial[u,x,emb]

gives the minimal polynomial of u relative to the finite field embedding emb.

# Details and Options • MinimalPolynomial[s,x] gives the lowest-degree polynomial with integer coefficients, positive leading coefficient and the GCD of all coefficients equal to for which the algebraic number s is a root.
• gives a pure function representation of the minimal polynomial of s.
• MinimalPolynomial[s,x,Extension->a] finds the characteristic polynomial of over the field .
• For a FiniteFieldElement object u in a finite field of characteristic , MinimalPolynomial[u, x] gives the lowest-degree monic polynomial with integer coefficients between and for which u is a root.
• MinimalPolynomial[u,x,k] gives the lowest-degree monic polynomial with coefficients from the -element subfield of for which u is a root. k needs to be a divisor of the extension degree of over .
• If emb=FiniteFieldEmbedding[e1e2], then MinimalPolynomial[u,x,emb] gives the polynomial with coefficients in the ambient field of e1 that map through emb to the coefficients of the minimal polynomial of u over the image of emb.

# Examples

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## Basic Examples(2)

Minimal polynomials of algebraic numbers:

Minimal polynomials of finite field elements:

## Scope(6)

### Algebraic Numbers(5)

Root objects:

AlgebraicNumber objects:

Pure function minimal polynomial:

### Finite Field Elements(1)

Represent a finite field with characteristic and extension degree :

Minimal polynomial over :

Minimal polynomial over with coefficients given as elements of :

Minimal polynomial over the -element subfield of :

Embed a field with elements in :

Minimal polynomial relative to the finite field embedding :

Pure function minimal polynomial:

## Options(1)

### Extension(1)

Find the characteristic polynomial of over the extension of :

The characteristic polynomial is a power of the minimal polynomial of :

## Applications(3)

Construct a polynomial with a root :

The degree of the number field generated by (2-I)/Sqrt:

Check whether a finite field element generates its ambient field:

## Properties & Relations(6)

Compute the extension that defines the number field :

Find the characteristic polynomial of over :

The characteristic polynomial is a power of the minimal polynomial of :

Use FrobeniusAutomorphism to find all conjugates of a finite field element a:

The conjugates are roots of the minimal polynomial of a:

If MinimalPolynomial[a,x]xn+cn-1xn-1++c0, then :

If MinimalPolynomial[a,x,k]xn+cn-1xn-1++c0, then :

If MinimalPolynomial[a,x]xn+cn-1xn-1++c0, then :

If MinimalPolynomial[a,x,k]xn+cn-1xn-1++c0, then :