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
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:
MinimalPolynomial[Sqrt[3], x]MinimalPolynomial[Sqrt[2] + Sqrt[3], x]Minimal polynomials of finite field elements:
ℱ = FiniteField[2, 12];MinimalPolynomial[ℱ[123], x]MinimalPolynomial[ℱ[123], x, 3]Scope (6)
Algebraic Numbers (5)
MinimalPolynomial[Sqrt[1 + Sqrt[3]], x]MinimalPolynomial[(1 + I) / Sqrt[2], x]Root objects:
MinimalPolynomial[Root[2#1 ^ 3 - 2 #1 + 7&, 1] + 17, x]AlgebraicNumber objects:
MinimalPolynomial[AlgebraicNumber[Root[-2 + #1 ^ 3&, 2], {2, 2, 1}], x]MinimalPolynomial automatically threads over lists:
MinimalPolynomial[{Sqrt[3], Root[-2 + #1 ^ 3&, 2] + 1}, x]Pure function minimal polynomial:
MinimalPolynomial[Sqrt[2] + Sqrt[3]]Finite Field Elements (1)
Represent a finite field with characteristic 
and extension degree 
:
ℱ = FiniteField[17, 6]
MinimalPolynomial[ℱ[123], x]Minimal polynomial over 
with coefficients given as elements of 
:
MinimalPolynomial[ℱ[123], x, 1]Minimal polynomial over the 
-element subfield of 
:
MinimalPolynomial[ℱ[123], x, 2]Embed a field 
with 
elements in 
:
𝒦 = FiniteField[17, 3];
ℰ = FiniteFieldEmbedding[𝒦, ℱ]Minimal polynomial relative to the finite field embedding 
:
MinimalPolynomial[ℱ[123], x, ℰ]Pure function minimal polynomial:
MinimalPolynomial[ℱ[123]]Options (1)
![TemplateBox[{}, Rationals][ⅇ^(ⅈ pi/4)]](Files/MinimalPolynomial.en/27.png "TemplateBox[{}, Rationals][ⅇ^(ⅈ pi/4)]"){kind=small}
Applications (3)
Construct a polynomial with a root 
:
MinimalPolynomial[Sqrt[1 + Sqrt[2]], x]Plot[%, {x, -2, 2}, Epilog -> {Blue, PointSize[Large], Point[{Sqrt[1 + Sqrt[2]], 0}]}]The degree of the number field generated by (2-I)/Sqrt[5]:
Exponent[MinimalPolynomial[(2 - I) / Sqrt[5], x], x]Check whether a finite field element generates its ambient field:
ℱ = FiniteField[3, 10];Exponent[MinimalPolynomial[ℱ[123], x], x] == Information[ℱ, "ExtensionDegree"]Exponent[MinimalPolynomial[ℱ[636], x], x] == Information[ℱ, "ExtensionDegree"]Properties & Relations (6)
Compute the extension that defines the number field 
:
F = ToNumberField[{Sqrt[3], E ^ (I Pi / 4)}, All][[1, 1]]Find the characteristic polynomial of 
over 
:
MinimalPolynomial[Sqrt[2] + Sqrt[3], x, Extension -> F]The characteristic polynomial is a power of the minimal polynomial of 
:
{Factor[%], MinimalPolynomial[Sqrt[2] + Sqrt[3], x]}Use FrobeniusAutomorphism to find all conjugates of a finite field element a:
ℱ = FiniteField[7, 5];
a = ℱ[123];
conj = Table[FrobeniusAutomorphism[a, k], {k, 0, 4}]The conjugates are roots of the minimal polynomial of a:
MinimalPolynomial[a] /@ conjIf MinimalPolynomial[a,x]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[149, 5];
a = ℱ[1234];
FiniteFieldElementTrace[a]MinimalPolynomial[a, x]Mod[-Coefficient[%, x, 4], 149]If MinimalPolynomial[a,x,k]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[43, 6];
a = ℱ[1234];
FiniteFieldElementTrace[a, 2]MinimalPolynomial[a, x, 2]-Coefficient[%, x, 2]If MinimalPolynomial[a,x]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[149, 5];
a = ℱ[1234];
FiniteFieldElementNorm[a]MinimalPolynomial[a, x]Mod[-Coefficient[%, x, 0], 149]If MinimalPolynomial[a,x,k]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[43, 6];
a = ℱ[1234];
FiniteFieldElementNorm[a, 2]MinimalPolynomial[a, x, 2]-Coefficient[%, x, 0]Text
Wolfram Research (2007), MinimalPolynomial, Wolfram Language function, https://reference.wolfram.com/language/ref/MinimalPolynomial.html (updated 2026).
CMS
Wolfram Language. 2007. "MinimalPolynomial." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2026. 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