 Solution

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.