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

By unique factorization of ideals in a number field, each nonzero integral ideal has a unique expressionwith only finitely many nonzero exponents. Since the ideal norm is multiplicative, formally expanding one geometric series for each prime ideal givesUniqueness of ideal factorization says that every term occurs exactly once. This proves the Euler product for the Dedekind zeta function as a formal identity.

For each rational prime $p$, defineThese are the three cases in splitting of rational primes in a quadratic field.

Set $x=p^{-s}$. In the split case the local factor of the Dedekind zeta function is $(1-x{)}^{-2}$; in the inert case it is $(1-x^2{)}^{-1}$ because the prime ideal has norm $p^2$; and in the ramified case it is $(1-x{)}^{-1}$. On the other hand, the product of the local factors of ${\zeta }_{\mathbb{Q}}(s)$ and $L(\chi ,s)$ iswhich gives exactly those three expressions when $\chi (p)=1,-1,0$. Multiplying over all rational primes proves the quadratic Dedekind zeta factorizationformally.