[[Field extension]]
# Normal extension

A [[field extension]] $L:K$ is **normal** iff every $f(x) \in K[x]$ has a root $\alpha \in L$ iff it splits into linear factors in $L[x]$. #m/def/field
Equivalently,[^2009] 

- Every embedding $L \hookrightarrow \overline{K}$ induces an [[Automorphism of a field extension|automorphism]] of $L:K$;
- $L$ is the [[Splitting field]] of a family of polynomials in $K[x]$.

  [^2009]: 2009\. [[Sources/@aluffiAlgebraChapter02009|Algebra: Chapter 0]], §VII.4.1, p. 431

> [!missing]- Proof
> #missing/proof

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