Theorem (Kronecker-Weber) \(K/\mathbb{Q}\) abelian \(\Leftrightarrow\) \(K\subset \mathbb{Q}(\zeta_m)\) for some \(m\). We can construct the maximal abelian extension \(\mathbb{Q}^{ab}\) of \(\mathbb{Q}\) = \(\mathbb{Q}(\{\zeta_m\}_m) = \mathbb{Q}(\overline{\mathbb{Q}}^\times_{tors})\). There is an analogue for imaginary quadratic fields with \(\overline{\mathbb{Q}}^\times\) replaced by \(E(\overline{\mathbb{Q}})\), the CM theory.
Fix a prime power \(q^k=m>2\). Let
\[K=\mathbb{Q}(\zeta_m)=\mathbb{Q}(\{\text{roots of }x^m-1\}) = \mathbb{Q}(\{\text{roots of }\Phi_m(x)).\]
Here for prime power it is easy to compute \(\Phi_m = \frac{x^{q^k}-1}{x^{q^{k-1}}-1}\) whose degree is \(\varphi(m)\). We have \(\mathrm{Gal}(K/\mathbb{Q})\cong (\mathbb{Z}/m\mathbb{Z})^\times\).
\(\Phi_m(x+1) = x^{\varphi(m)} + (\text{divisible by }q) + q\), this proves that \(\Phi_m\) is irreducible by Eisenstein’s criterion.
\((q) = (1-\zeta_m)^{\varphi(m)}\) in \(\mathcal{O}_K \Rightarrow\) \(q\) is totally ramified in \(K/\mathbb{Q}\).
Observed by setting \(x=\zeta_m\) in $_m=
All other primes \(p\nmid \Delta_{x^m-1}=\pm m^m\) are unramified in \(K/\mathbb{Q}\).
\(\mathrm{Gal}(\mathbb{Q}(\zeta_{q^k})/\mathbb{Q}) = (\mathbb{Z}/q^k\mathbb{Z})^\times\), mapping \(i\mapsto (\zeta_m\mapsto \zeta_m^i)\).
\(\mathrm{Gal}(\bigcup \mathbb{Q}(\zeta_{q^k})/\mathbb{Q}) = \mathbb{Z}_q^\times\) (the \(q\)-adic units).
Here \(K=\mathbb{Q}(\zeta_m)\) is the compositum of \(\mathbb{Q}(\zeta_{q^k})\) for all prime powers \(q^k|| m\). We have
It has degree \(\prod \varphi(q^k) = \varphi(m)\).
\(p\nmid m\Rightarrow p\) is unramified in \(K/\mathbb{Q}\).
the residue degree \(f_p = [\mathbb{F}_p(\zeta_m):\mathbb{F}_p]\) which is the smallest \(r\) such that \(p^r\equiv 1\mod m\), or order of \(p\) in \((\mathbb{Z}/m\mathbb{Z})^\times\).
\(p|m=p^km_0\Rightarrow p\) is ramified in \(K/\mathbb{Q}\) with ramification degree \(\varphi(p^k)\) with residue degree the order of \(p\) in \((\mathbb{Z}/m_0\mathbb{Z})^\times\).
\(\mathrm{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q}) = (\mathbb{Z}/m\mathbb{Z})^\times\), so
\[\mathrm{Gal}(\mathbb{Q}^{ab}/\mathbb{Q}) = \varprojlim (\mathbb{Z}/m\mathbb{Z})^\times = \widehat{\mathbb{Z}}^\times.\]
Theorem.
\[\zeta_{\mathbb{Q}(\zeta_m)} = \prod_{\chi:(\mathbb{Z}/m\mathbb{Z})^\times\to \mathbb{C}^\times} L(s,\chi) = \prod_{p\nmid m} \frac{1}{1-p^{-s}}\prod_{p|m} \frac{1}{1-p^{-s}}\prod_{\chi:(\mathbb{Z}/m\mathbb{Z})^\times\to \mathbb{C}^\times} L(s,\chi).\]
Proof. Compute local factors \(F_p(T)\) at every prime \(p\), say \(m=p^km_0\). Let \(f\) be the order of \(p\) in \((\mathbb{Z}/m_0\mathbb{Z})^\times\), \(e = \varphi(p^k)\). Then \(g = \frac{\varphi(m)}{ef} = \frac{\varphi(m_0)}{f}\).
\[\begin{align*} F_p(T) &= (1-T^f)^r \\ &= \prod_{\chi} (1-\chi(p)T) \end{align*}\] Its degree is \(\varphi(m_0)\) and roots are \(f\)-th roots of unity. (characters whose modulus \(\nmid m_0\) evaluates \(\chi(p)=0\).)
Corollary. \(L(\chi,1)\neq 0\) for \(\chi\neq 1\).
Proof.
\[\zeta_{\mathbb{Q}(\zeta_m)} = \zeta(s)\cdot \prod_{\chi\neq 1} L(s,\chi).\]
the left hand side and \(\zeta(s)\) have a simple pole at \(s=1\), so the product have no poles and cannot be \(0\) at \(s=1\).
The Dirichlet’s theorem of primes on arithmetic progressions says
\[\sum_{p\equiv a\mod q} \frac{1}{p^s} = \frac{1}{\varphi(m)}\sum_\chi \overline{\chi(a)} \log L(\chi,s) + \text{terms analytic at }s=1.\]
In general \(K/\mathbb{Q}\) abelian \(\Rightarrow\) \(\zeta_K(s)=\prod \{L-functions of Dirichlet charcters\}\) (need representation theory, later).
Example. \(m=12\), \(\mathbb{Q}(\zeta_m) = \mathbb{Q}(i,\sqrt{-3})\) biquadratic. If we lookat characters
\[{(\mathbb{Z}/12\mathbb{Z})^\times} = \{1,\chi_3,\chi_4,\chi_12\}.\]
\[L(1,s) \]
\[L(\chi_3,s)\]
\[L(\chi_4,s)\]
\[L(\chi_12,s)\]