normal extension

《代数学引论》 · 第 215 页 · 定义 3

一个代数扩张 $K/F$ 叫做正规扩张，如果 $K/F$ 具有下列的性质：若 $F[x]$ 的任一个不可约多项式在 $K$ 内有一根，则它在 $K$ 内可以完全分解成一次因式的乘积。

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《A Course in Finite Group Representation Theory》 · 第 108 页 · Section 3.5「Splitting Fields」，Theorem 4 and Definition 5

Theorem 4. The following conditions are equivalent for a field $K$ and an algebraic extension $K\subset L$: (i) Every $K$-homomorphism $L\to\overline L$ into an algebraic closure $\overline L$ of $L$ restricts to an automorphism of $L$. (ii) $L$ is a splitting field of a family of polynomials in $K[X]$. (iii) Every irreducible polynomial in $K[X]$ that admits a zero in $L$ decomposes over $L$ completely into linear factors. Definition 5. An algebraic field extension $K\subset L$ is called normal if it satisfies the equivalent conditions of Theorem 4.