 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-1/20g/b/solution

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

The trace/norm integrality conditions give the stated congruences for

θ = (a + bi + c m + d i m) /2

. Solving them yields these integral bases:

m \equiv 1 (mod 4) : 1, i, \frac{1 + m}{2}, \frac{i + i m}{2},

m \equiv 3 (mod 4) : 1, i, \frac{i + m}{2}, \frac{1 + i m}{2} .

The three quadratic subfields have fundamental discriminants

- 4, D_{m}, D_{- m}

D_{L} = (- 4) D_{m} D_{- m} = 16 m^{2}

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