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.
  • MinimalPolynomial[s] 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)

Radical expressions:

Root objects:

AlgebraicNumber objects:

MinimalPolynomial automatically threads over lists:

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 TemplateBox[{}, Rationals][ⅇ^(ⅈ pi/4)] 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[5]:

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 :

Wolfram Research (2007), MinimalPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/MinimalPolynomial.html (updated 2023).

Text

Wolfram Research (2007), MinimalPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/MinimalPolynomial.html (updated 2023).

CMS

Wolfram Language. 2007. "MinimalPolynomial." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/MinimalPolynomial.html.

APA

Wolfram Language. (2007). MinimalPolynomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MinimalPolynomial.html

BibTeX

@misc{reference.wolfram_2024_minimalpolynomial, author="Wolfram Research", title="{MinimalPolynomial}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/MinimalPolynomial.html}", note=[Accessed: 23-May-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_minimalpolynomial, organization={Wolfram Research}, title={MinimalPolynomial}, year={2023}, url={https://reference.wolfram.com/language/ref/MinimalPolynomial.html}, note=[Accessed: 23-May-2024 ]}