Normal Extension

definitionfield-theory

Definition

For ${}^E{/}_k$ algebraic (not necessarily finite), we say ${}^E{/}_k$ is a normal extension if for every $\alpha \in E$, $E$ contains a copy of ${\mathrm{split}}_k{m}_{\alpha /k}(x)$.

Equivalently, $\alpha \in E$ implies $m_{\alpha /k}(x)$ factors $m_{\alpha /k}(x)=(x-{\alpha }_1)\cdots (x-{\alpha }_{\mathrm{deg}(\frac{\alpha }{k})})$ for some ${\alpha }_i\in E$ that may repeat.

Examples

Any $[E:k]\le 2$ is normal.

If $[E:k]=1⟹E=k\forall \alpha \in E$, $m_{\alpha /k}(x)=x-\alpha$ has “all” roots in $E=k$. If $[E:k]=2⟹\left\{1,\alpha \right\}$ is a basis $\forall \alpha \in E\setminus k$. $\mathrm{deg}{m}_{\alpha /k}(x)=2$ and $m_{\alpha /k}(x)=(x-\alpha )(x-\beta )$ ($\alpha$ is a root of $m_{\alpha /k}(x)$ and $\beta$ is necessarily also in $E$

If $m_{\alpha /k}(x)=x^2+ax+b$ for $a,b\in k$ $⟹$ $\alpha +\beta =-a⟹\beta =-a-\alpha \in E$.