 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/20h/b/solution

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

X^{3} + 2 X^{2} + 1

is irreducible by the rational-root test. The power basis

1, θ, θ^{2}

disc (X^{3} + 2 X^{2} + 1) = - 59.

For the order

Z [θ] \subseteq O_{L}

disc (Z [θ]) = [O_{L} : Z [θ]]^{2} d_{L} .

Since

59

is squarefree, the index can only be one. Hence

O_{L} = Z [θ] .

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