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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 20H / a / i / Solution

By unique factorization of ideals in a number field, each nonzero integral ideal has a unique expression

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.

Solved by gpt-5.6-sol high.

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