Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3

A Dirichlet character modulo $m$ is a homomorphismExtend it periodically to $\mathbb{Z}$ by setting $\chi (n)=0$ when $(n,m)>1$. Its Dirichlet L-function is

The Dedekind zeta function of $K$ isfor $\mathrm{Re}s>1$.

By the Kronecker–Weber theorem, choose $m$ withRestriction gives a quotientEvery character $\psi \in \hat{G}$ inflates along this quotient and has an associated primitive Dirichlet character ${\chi }_{\psi }$, whose conductor may divide $m$. Comparing Euler factors, or applying the factorization of the Artin $L$-function of the regular representation, gives

The group $(\mathbb{Z}/7\mathbb{Z}{)}^{\times }$ is cyclic of order six, generated by $3$. Put $\omega =e^{2\pi i/6}$. The six characters areand vanish on multiples of $7$. Sincetheir values are explicitlyTaking $j=0,\dots ,5$ lists all six characters.

The unique order-two character is ${\chi }_3$, the Legendre symbolIt is $1$ on $1,2,4$, is $-1$ on $3,5,6$, and is zero on multiples of $7$. Since the unique quadratic subfield of $\mathbb{Q}({\zeta }_7)$ is $\mathbb{Q}(\sqrt{-7})$, this is the character corresponding to $K$.

For $K=\mathbb{Q}(\sqrt{-7})$,The character is odd, ${\chi }_3(2)=1$, andThe supplied odd-character formula gives

The analytic class number formula for this imaginary quadratic field isbecause $w_K=2$ and $d_K=-7$. Since ${\mathrm{Res}}_{s=1}\zeta (s)=1$, comparison with $L(1,{\chi }_3)$ yields $\boxed{h_K=1}$.