 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/18h/c/solution

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

Because

lcm (9, 15) = 45

and suitable integral powers of

ζ_{9}

and

ζ_{15}

generate a primitive forty-fifth root,

L_{1} L_{2} = Q (ζ_{45}) .

The degree formula gives

[L_{1} \cap L_{2} : Q] = \frac{φ ( 9 ) φ ( 15 )}{φ ( 45 )} = \frac{6 \cdot 8}{24} = 2.

Both fields contain

Q (ζ_{3})

, which already has degree two, so

L_{1} \cap L_{2} = Q (ζ_{3}) .

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