MinimalPolynomial

The minimal polynomial of the algebraic number $a$ in $x$: the irreducible polynomial over $\mathbb{Z}$ with positive leading coefficient that $a$ is a root of.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMinimalPolynomial✓↗

MinimalPolynomial(a, x)the minimal polynomial of $a$, for $a$ written in one radical of a rational, $i$, the golden ratio or one PolynomialRoot

Typed signatures

Type	Package	Applies
(number, any) -> expression	number-theory	always

Domain: Number theory

Details

• $a$ is read as a polynomial in one generator: $1 + \sqrt[3]{2}$, $(1 + \sqrt{-3})/2$ or $\theta^2 - \theta$ for $\theta$ a PolynomialRoot; with two generators ($\sqrt 2 + \sqrt 3$) it stays unevaluated
• Its degree is $[\mathbb{Q}(a) : \mathbb{Q}]$; $a$ is an algebraic integer exactly when the polynomial is monic (AlgebraicIntegerQ)

Examples

Basic #

$(x - 1)^3 = 2$

live ↗

In$\mathrm{MinimalPolynomial}(1+\sqrt[3]{2}, x)$Out$x^3-3x^2+3x-3$

$\varphi^2 = \varphi + 1$

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In$\mathrm{MinimalPolynomial}(\varphi, x)$Out$x^2-x-1$

$\sqrt[3]{4}/2$ satisfies $2x^3 = 1$: not monic, so not an algebraic integer

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In$\mathrm{MinimalPolynomial}(\frac{2^{\frac{2}{3}}}{2}, x)$Out$2x^3-1$

$(1 + \sqrt{-3})/2$, a sixth root of unity

live ↗

In$\mathrm{MinimalPolynomial}(\frac{1+\sqrt{-3}}{2}, x)$Out$x^2-x+1$

About

Examples	4
Checked against	Wolfram Language 4 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/ce-patches/src/compute-engine/library/algebraic-numbers.ts

mappedwolframpackages/oracle/src/mappings.ts

See also: AlgebraicIntegerQ, AlgebraicNumberNorm, PolynomialRoot