\(K/\mathbb{Q}\) be finite Galois, \(p\) a prime of \(\mathbb{Q}\). \(G=\mathrm{Gal}(K/\mathbb{Q})\) and \(|G|=[K:\mathbb{Q}]=d\).
\(\mathfrak{p}_1,\ldots,\mathfrak{p}_r\) primes of \(K\) above \(p\).
They have the same ramification index \(e\) and residue class degree \(f\) since \(K/\mathbb{Q}\) is Galois.
\(G\) acts \(\{\mathfrak{p}_i\}\) transitively.
Define \(D_{\mathfrak{p}_i} = \mathrm{Stab}_G(\mathfrak{p}_i)\)
\[D_{\mathfrak{p}_i} = \{\sigma\in G\mid \sigma(\mathfrak{p}_i)=\mathfrak{p}_i\}\]
It acts on \(\mathcal{O}_K/\mathfrak{p}_i\cong \mathbb{F}_{p^f}\). The reduction map is
\[D_{\mathfrak{p}_i}\to \mathrm{Gal}(\mathbb{F}_{p^f}/\mathbb{F}_p) = \langle x\mapsto x^p\rangle \cong C_f\]
This reduction map is onto.
Define \(I_{\mathfrak{p}_i} = \ker(D_{\mathfrak{p}_i}\to C_f)\).
\(\mathrm{Frob}_{\mathfrak{p}_i}\) is any element \(\sigma\in D_{\mathfrak{p}_i}\) such that \(\sigma(x)\equiv x^p\pmod{\mathfrak{p}_i}\).
Let \(K/\mathbb{Q}\) be Galois, \(p\) a prime of \(\mathbb{Q}\).
\(D_p := D_{\mathfrak{p}_i}\) for \(\mathfrak{p}_i|p\), which is only determined up to conjugation.
\(I_p := I_{\mathfrak{p}_i}\)
\(\mathrm{Frob}_p := \mathrm{Frob}_{\mathfrak{p}_i}\)
Theorem (Prime Decomposition) Let \(K/\mathbb{Q}\) be Galois, \(p\) a prime of \(\mathbb{Q}\). \(\mathbb{Q}\subset F\subset K\)
\[ G = \mathrm{Gal}(K/\mathbb{Q}), \quad H=\mathrm{Gal}(K/F), \quad F=K^H \]
If \(p=\mathfrak{p}^{e_1}\dots \mathfrak{p}_r^{e_r}\) is the decomposition in \(F\), \(D\) the decomposition group at \(p\), \(I\) the inertia group at \(p\) normal inside \(D\).
Then
\[\begin{align*} \{\mathfrak{p}_i\} & \Leftrightarrow \{\text{Double cosets } DgH\in D\backslash G/H\} \\ & = \{\text{Orbits of } D\text{ on the }G\text{-set } G/H\} \end{align*}\]
each orbit has length \(e_if_i\) and is a union of \(f_j\) \(I\)-orbits of length \(e_j\), cyclically permuted by \(\mathrm{Frob}_p\).
Proof. Let \(F=\mathbb{Q}[x]/(f)\) and \(p\nmid \Delta_f\), \(\mathfrak{P}|p\) in \(F\), assume \(I=\{1\}\).
\[\begin{align*} G/H &\xLeftrightarrow{G\text{-set}} \{\text{roots of }f\in \overline{\mathbb{Q}}\} \\ &\xLeftrightarrow{D\text{-set}} \{\text{roots of }f\text{ mod }p\text{ in }\overline{\mathbb{F}_p}\} \\ \end{align*}\]
Observe Frobenius action on both sides and examine their orbits.
When \(p\mid \Delta_f\) use local fields instead of Kummer-Dedekind.
Theorem(Chebotarev) Let \(K/\mathbb{Q}\) be Galois, \(G=\mathrm{Gal}(K/\mathbb{Q})\), \(H\subset G\) a subgroup.
\(\mathrm{Frob}_p\in G\) is equi-distributed in the following sense, for any conjugacy class \(C\subset G\), the density of primes \(p\) such that \(\mathrm{Frob}_p\in C\) is \(|C|/|G|\). (Does this mean there is a natural measure on something?)
Example \(F=\mathbb{Q}(2^{1/3}), K=\mathbb{Q}(\zeta_3,2^{1/3})\), \(G=\mathrm{Gal}(K/\mathbb{Q})\), \(H=\mathrm{Gal}(K/F)\).
\(G=\mathrm{Gal}(K/\mathbb{Q})=S_3\), generated by the rotation \(\sigma = (123) : 2^{1/3}\mapsto 2^{1/3}\zeta_3\), and complex conjugation \(\tau=(23) : \zeta_3\mapsto \zeta_3^{-1}\).
All primes \(p\neq 2,3\) are unramified in \(K/\mathbb{Q}\) which implies \(I_p=1\), \(D_p=\langle \mathrm{Frob}_p\rangle\) is cyclic.
The action of \(D_p\) on \(G/H\)
\[G/H = \{1,(23)\}\sqcup \{(123),(13)\}\sqcup \{(132),(12)\}\]
\(\mathrm{Frob}=\mathrm{id}\), \(D_p=\langle \mathrm{Frob}_p\rangle=\{1\}\), \(D\) has trivial action on \(G/H\).
\(f_1=f_2=f_3=1\)
\(\mathrm{Frob}_p=\tau\) is a \(2\)-cycle (chosen up to conjugation), the orbits of action of \(D_p\) on \(G/H\) is
\[\{H\}\sqcup \{(123)H, (132)H\}\]
\(f_1=1,f_2=2\).
\(\mathrm{Frob}_p=\sigma\) is a \(3\)-cycle, the orbits of action of \(D_p\) on \(G/H\) is
\[\{(123)H,(13)H,(12)H\}\]
\(f_1=3\).
Primes \(p=2,3\) are ramified in \(K/\mathbb{Q}\).
Let \(G\) be a finite group, a \(d\) dimensional complex representations of \(G\) is a group homomorphism \(\rho:G\to \mathrm{GL}(V)\cong \mathrm{GL}_d(\mathbb{C})\) where \(V\) is a \(d\) dimensional complex vector space.
When \(K/F\) is a finite Galois extension of number fields, the representation
\[\rho: \mathrm{Gal}(K/F) \to \mathrm{GL}(V)\]
is called an Artin representation over \(F\).
For example a one dimensional Artin representation is a character \(\rho: \mathrm{Gal}(K/F)\to \mathbb{C}^\times\).
In case of a profinite group
\[\rho: \mathrm{Gal}(\overline{F}/F)\to \mathbb{C}^\times\]
we require the continuity of \(\rho\), i.e. we can make the representation to differ on two elements arbitrarily small by choosing a finite Galois extension \(K/F\), on which \(\mathrm{Gal}(K/F)\) two elements of \(\mathrm{Gal}(\overline{F}/F)\) agree.