 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/18j/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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.

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