Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 20G / b

Table of contents
- Solution b

Solution

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}

. By discriminant of a biquadratic field,

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

in both residue classes.

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