Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 20H / a / i / Solution Created 2026-09-24 Updated 2026-09-25
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By unique factorization of ideals in a number field, each nonzero integral ideal has a unique expression
(1)
with only finitely many nonzero exponents. Since the ideal norm is multiplicative, formally expanding one geometric series for each prime ideal gives
(2)
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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