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.