 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-2/18f/c/solution

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

Identify

G ≃ (Z /15 Z)^{\times} ≃ C_{4} \times C_{2}

, with

σ (ζ) = ζ^{2}

and

τ (ζ) = ζ^{- 1}

. Its subgroups are

1; ⟨ σ^{2} ⟩, ⟨ τ ⟩, ⟨ σ^{2} τ ⟩; ⟨ σ ⟩, ⟨ σ τ ⟩, ⟨ σ^{2}, τ ⟩; G .

Their fixed fields, in the same layers, are

L; Q (- 3, 5), L^{+} = Q (ζ + ζ^{- 1}), Q (ζ_{5}); Q (- 15), Q (- 3), Q (5); Q .

The real fields are

Q

Q (5)

, and

L^{+}

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