Minimal polynomial of an element in an extension field

Overview and basic definition

Minimal polynomial

If $E$ is an extension field of $F$ and $\alpha \in E$ is algebraic over $F$, then the minimal polynomial $\text{irr}(\alpha ,F)$ is the unique irreducible monic polynomial that generates $\ker {\text{ev}}_{\alpha }$. By definition, $\text{irr}(\alpha ,F)$ is the polynomial of smallest degree for which $\alpha$ is a root (i.e., for all $g\in F[x]$, we have $g(\alpha )=0$ iff $\text{irr}(\alpha ,F)$ divides $g$).

The value of $\mathrm{deg}\text{irr}(\alpha ,F)$ is called the degree of $\alpha$ over $F$.

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Properties

• Extension fields: In the case that $E=F(\alpha )$ is a simple extension of $F$ with minimal polynomial $f=\text{irr}(\alpha ,F)$, the kernel of the evaluation homomorphism ${\text{ev}}_{\alpha }:F[x]\to F(\alpha ),g\mapsto g(\alpha )$ is precisely the principal ideal generated by $f$: $\ker ({\text{ev}}_{\alpha })=\{g\in F[x]:g(\alpha )=0\}=(f).$ Hence by (Theorem) First isomorphism theorem, ${\text{ev}}_{\alpha }$ induces an isomorphism ${\overline{\text{ev}}}_{\alpha }:F[x]/(f)\to F(\alpha ),g+(f)\mapsto g(\alpha ),$where the inverse mapping is defined on the generator by $\alpha \mapsto x+(f)$, and otherwise $a\mapsto a+(f)$ for $a\in F$.

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Notes

Minimal polynomial

If $E$ is an extension field of $F$ and $\alpha \in E$ is algebraic over $F$, then there exists a unique irreducible monic polynomial $p\in F[x]$ that generates $\ker {\text{ev}}_{\alpha }=(p)$, which is denoted $\text{irr}(\alpha ,F)$.

Facts for finding $\text{irr}(\alpha ,F)$

Let $E$ be an extension field of $F$ and $\alpha \in E$.

• (i) If $p\in F[x]$ is any irreducible monic polynomial for which $\alpha$ is a root, i.e., $p(\alpha )=0$, then $p=\text{irr}(\alpha ,F)$ is unique.
• (ii) Suppose $K$ is an extension field of $E$ and $\beta \in K$. If $\beta$ is algebraic over $F$, then $\beta$ is algebraic over $E$ (by definition) and $\text{irr}(\beta ,E)$ divides $\text{irr}(\beta ,F)$.