 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

The equality of residue-field sizes says that the residue degree

f (Q / P)

is one. Since

Q = α O_{L}

, the principal ideal factorization of

α

has valuation one at

Q

and zero at every other prime. Taking the relative ideal norm gives

N_{L / K} (α) O_{K} = N_{L / K} (α O_{L}) = N_{L / K} (Q) = P^{f (Q / P)} = P .

Thus $P = N_{L / K} (α) O_{K}$ .

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