Field extension

Normal extension

A field extension $𝐿 : 𝐾$ is normal iff every $𝑓 (𝑥) \in 𝐾 [𝑥]$ has a root $𝛼 \in 𝐿$ iff it splits into linear factors in $𝐿 [𝑥]$ . #m/def/field Equivalently,¹

Every embedding $𝐿 ↪ ―― 𝐾$ induces an automorphism of $𝐿 : 𝐾$ ;
$𝐿$ is the Splitting field of a family of polynomials in $𝐾 [𝑥]$ .

Proof

#missing/proof

#state/develop | #lang/en | #SemBr

2009. Algebra: Chapter 0, §VII.4.1, p. 431 ↩