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    Characterizations of finite Galois extensions

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    Finite Galois means separable splitting, fixed field $F$, full automorphism count, or normal plus separable.Suggest a slogan

    Let \(K/F\) be a finite field extension. The following are equivalent:

    1. \(K\) is a splitting field over \(F\) of a separable polynomial in \(F[x]\).

    2. The fixed field of \(\Aut(K/F)\) is \(F\).

    3. \([K:F]=\abs{\Aut(K/F)}\).

    4. \(K/F\) is normal and separable.

    A finite extension satisfying these conditions is Galois.

    What four conditions each characterise a Galois extension $K/F$?

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