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 $f \in K \left[{x}\right]$ such that $f \left({\alpha}\right) = 0$.
Definition 3
The minimal polynomial of $\alpha$ over $K$ is the unique monic polynomial $f \in K \left[{x}\right]$ that generates the kernel of the evaluation homomorphism $K \left[{x}\right] \to L$ at $\alpha$.
That is, such that for all $g \in K \left[{x}\right]$:
- $g \left({\alpha}\right) = 0$ 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 $\map {\mu_K} \alpha$.
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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 $\index {\map \Q \theta} \Q = 4$.