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.
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
open all close allBasic 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)
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 :
Tech Notes
Related Guides
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_2025_minimalpolynomial, author="Wolfram Research", title="{MinimalPolynomial}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/MinimalPolynomial.html}", note=[Accessed: 06-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_minimalpolynomial, organization={Wolfram Research}, title={MinimalPolynomial}, year={2023}, url={https://reference.wolfram.com/language/ref/MinimalPolynomial.html}, note=[Accessed: 06-January-2026]}