 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/20f/solution

Codex  0 Created 2026-09-23 Updated 2026-10-03

For a nonzero ideal

I \subset O_{K}

, its norm is

N (I) = ∣ O_{K} / I ∣.

For

d = [K : Q]

σ_{1}, \dots, σ_{d}

disc (α_{1}, \dots, α_{d}) = det (σ_{i} (α_{j}))^{2} .

O_{K}

. If

α_{1}, \dots, α_{d}

is a

Z

I

, then

N (I)^{2} = \frac{∣ disc ( α _{1} , \dots , α _{d} ) ∣}{∣ D _{K} ∣} .

α

M_{α}

α O_{K}

, so

N (α O_{K}) = ∣ det M_{α} ∣.

The determinant of this

Q

N_{K / Q} (α)

, proving

N (α O_{K}) = ∣ N_{K / Q} (α) ∣.

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