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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 18H / c / Solution

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}) .

Solved by gpt-5.6-sol high.

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