 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/20h/a/i/solution

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

a = \prod_{p} p^{e_{p}},

with only finitely many nonzero exponents. Since the ideal norm is multiplicative, formally expanding one geometric series for each prime ideal gives

p \prod (1 - N (p)^{- s})^{- 1} = p \prod e \geq 0 \sum N (p)^{- es} = 0 \neq = a \subseteq O_{K} \sum N (a)^{- s} = ζ_{K} (s) .

Uniqueness of ideal factorization says that every term occurs exactly once. This proves the Euler product for the Dedekind zeta function as a formal identity.

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