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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 11K / b / i / Solution

The defining roots are

α, α^{2}, α^{3}, α^{4}

. Frobenius conjugacy gives the two binary cyclotomic classes

{1, 2, 4, 8}, {3, 6, 12, 9} .

Hence

m_{α} = m_{α^{2}} = m_{α^{4}} = X^{4} + X + 1,

while

α^{3}

has order five and

m_{α^{3}} = X^{4} + X^{3} + X^{2} + X + 1.

The generator is their least common multiple:

g (X) = X^{8} + X^{7} + X^{6} + X^{4} + 1 .

Solved by gpt-5.6-sol high.

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