---
tag: 0GK1WB9R98
kind: definition
name: Minimal polynomial of an algebraic element
msc: [12F05]
defines: 0VX9EP2GS8
uses: [0NPW1DWECM, 0BDX4MBKT4, 0DPF6WRZW6]
sources:
  - url: https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory)
    locator: Definition
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be [[algebraic|0NPW1DWECM]] over $K$.
The minimal polynomial of $a$ over $K$ is the [[monic polynomial|0BDX4MBKT4]] $m_a \in K[x]$ of smallest [[degree|0DPF6WRZW6]] such that $m_a(a) = 0$.

## Remarks

The minimal polynomial exists and is unique. If two monic polynomials of the smallest degree vanish at $a$, their difference has smaller degree and vanishes at $a$, so it is zero.
The minimal polynomial [[is irreducible|0NFEG48RDR]], and it [[divides each polynomial that vanishes at $a$|06K7SARZKR]].
The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$ is $x^2 - 2$. Over $\mathbb{R}$, it is $x - \sqrt{2}$.
The minimal polynomial of $i$ over $\mathbb{R}$ is $x^2 + 1$.

## Explanations

### L1 human

Among all the polynomial equations that an algebraic element satisfies, the minimal polynomial is the simplest one.
It depends on the base field: a bigger base field can give a smaller minimal polynomial.