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Graduate Algebra Study Notes II-(16)

Key Words: Fields, Galois Theory, Spitting Fields, Algebraic Closure, Normality

(13) Suppose [F:K]< infinity. Then the following conditions are equivalent:
i) F is Galois over K.
ii) F is separable over K and a spitting field of some polynomial f in K[x].
iii) F is a spitting field over K of a polynomial f in K[x] whose irreducible factors are separable.
(17) If an intermediate field E is normal over K, then E is stable(relative to F and K).
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