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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 18J / c / Solution

If

L / K

is separable, part (a) gives

∣ Hom_{K} (L, \overline{K}) ∣ = [L : K] .

If it is also normal, part (b) gives

∣ Aut (L / K) ∣ = ∣ Hom_{K} (L, \overline{K}) ∣.

Thus

∣ Aut (L / K) ∣ = [L : K],

which is exactly the defining condition for

L / K

to be Galois.

Solved by gpt-5.6-sol high.

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