Field extension

Normal extension

A field extension 𝐿 :𝐾 is normal iff every 𝑓(𝑥) ∈𝐾[𝑥] has a root 𝛼 ∈𝐿 iff it splits into linear factors in 𝐿[𝑥]. 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

proof

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develop | en | SemBr

Footnotes

1. 2009. Algebra: Chapter 0, §VII.4.1, p. 431 ↩