# Definition:Minimal Polynomial

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## Definition

Let $L / K$ be a field extension.

Let $\alpha \in L$ be algebraic over $K$.

### Definition 1

The **minimal polynomial** of $\alpha$ over $K$ is the unique monic polynomial $f \in K \sqbrk x$ of smallest degree such that $\map f \alpha = 0$.

### Definition 2

The **minimal polynomial** of $\alpha$ over $K$ is the unique irreducible, monic polynomial $f \in K \sqbrk x$ such that $\map f \alpha = 0$.

### Definition 3

The **minimal polynomial** of $\alpha$ over $K$ is the unique monic polynomial $f \in K \sqbrk x$ that generates the kernel of the evaluation homomorphism $K \sqbrk x \to L$ at $\alpha$.

That is, such that for all $g \in K \sqbrk x$:

- $\map g \alpha = 0$ if and only if $f$ divides $g$.

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Note that it depends only on $\alpha$ and $K$.

## Also known as

A **minimal polynomial** is also seen referred to as a **minimum polynomial**.

The **minimal polynomial** of $\alpha$ over $K$ can be denoted $\map {\min_K} \alpha$ or $\map {\mu_K} \alpha$.

## Examples

### Minimal Polynomial for $\sqrt 2$

The minimal polynomial of $\sqrt 2$ in $\Q$ is $x^2 - 2$.

### Minimal Polynomial for $\sqrt {2 + \sqrt 3}$

Let $\theta = \sqrt {2 + \sqrt 3}$.

The minimal polynomial of $\theta$ in $\Q$ is $x^4 - 4 x + 1$.

Hence $\index {\map \Q \theta} \Q = 4$.

## Also see

- Equivalence of Definitions of Minimal Polynomial
- Minimal Polynomial Exists
- Minimal Polynomial is Unique

- Results about
**minimal polynomials**can be found**here**.