# Definition:Minimal Polynomial

## 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 $g$.

## 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 $\map {\mu_K} \alpha$.

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

## Examples

### 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

• Results about minimal polynomials can be found here.