How to determine the density of the set of completely splitting primes for a finite extension?

Solution 1:

Preliminary. Given a group $G$, a set equipped with an action of $G$ we call a $G$-set. A morphism $\varphi:X\to Y$ of two $G$-sets $X$ and $Y$ is a set-theoretic map that intertwines with the $G$-action, also known as a $G$-equivariant map, or $G$-map. This means that $\varphi(gx)=g\varphi(x)$ for all $x\in X$. If $f$ is a a $G$-equivariant bijection then we say it is an isomorphism of $G$-sets.

Let $X$ be a $G$-set. Recall that an orbit is a subset of $X$ of the form $Gx=\{gx:g\in G\}$ and a stabilizer is a subgroup of $G$ of the form $G_x=\{g\in G:gx=g\}$. If $H$ is a subgroup of $G$, not necessarily normal, the left coset space $G/H$ inherits a $G$-action.

Orbit-stabilizer theorem. The usual quantitative version says $|Gx|=[G:G_x]$. A qualitative version of the theorem says $aG_x\mapsto ax$ is an isomorphism $G/G_x\cong Gx$ of $G$-sets.

Preliminary. Recall that if $L/K$ is a separable extension of global fields and $\frak p$ a prime of ${\frak O}_K$ then it will factor in ${\frak O}_L$ as ${\frak pO}_L={\frak P}_1^{e_1}\cdots{\frak P}_g^{e_g}$ for some primes ${\frak P}_i$ of ${\frak O}_L$ and exponents $e_i$. These exponents are called ramification indices. The residue field ${\frak O}_K/{\frak p}$ embeds into the residue fields ${\frak O}_L/{\frak P}_i$ for each $i$, and the corresponding degrees are called the residue degrees. The ramification and residue data satisfy the relation $\sum_{i=1}^ge_if_i=[L:K]$. If $e_i>1$ for any $i$ we say $\frak p$ ramifies in ${\frak O}_L$. A prime of ${\frak O}_K$ ramifies in ${\frak O}_L$ if and only if it divides $\Delta_{L/K}$, the relative discriminant. If $L/K$ is Galois, $G_{L/K}$ acts transitively on $\{{\frak P}_1,\cdots,{\frak P}_g\}$. A stabilizer ${\rm Stab}({\frak P})$ is called the decomposition group $D_{{\frak P}\mid{\frak p}}$. The decomposition group of $\frak P\mid p$ acts on the residue field ${\frak O}_L/{\frak P}$, which induces a surjective map $D_{{\frak P}\mid{\frak p}}\to G_{l/k}$ (with $l={\frak O}_L/{\frak P}$, $k={\frak O}_K/{\frak p}$). Its kernel is the inertia group $I_{{\frak P}\mid{\frak p}}$.

If $M/L/K$ and ${\frak P}\mid\wp\mid{\frak p}$ then the ramification indices (indeed, the prime factorization) and residue degrees refine as $e({\frak P}|{\frak p})=e({\frak P}|\wp)e(\wp|{\frak p})$ and $f({\frak P}|{\frak p})=f({\frak P}|\wp)f(\wp|{\frak p})$. The original data back in $L/K$ satisfy the relation $e_1f_1+\cdots+e_gf_g=[L:K]$. If $L/K$ is Galois, by orbit-stabilizer we know $g=[G_{L/K}:D_{{\frak P}\mid{\frak p}}]$, and $e_1=\cdots=e_g=1$, $f_1=\cdots=f_g=f$, so $|G_{L/K}|=[L:K]=$ $\sum ef=gef$, and $|G_{l/k}|=[l:k]=f$, hence $|D_{{\frak P}\mid{\frak p}}|=ef$ and $|I_{{\frak P}\mid{\frak p}}|=e$. If $\frak p$ is unramified in $L$ then the inertia group is trivial and $D_{{\frak P}\mid{\frak p}}\cong G_{l/k}$.

The residue Galois group $G_{l/k}\cong C_f$ is pointed; it comes equipped with a canonical generator, the Frobenius automorphism $x\mapsto x^{|k|}$. The corresponding preimage of this map in $D_{{\frak P}\mid{\frak p}}$ (assuming $\frak p$ is unramified, else it'd have to be a coset of $I_{{\frak P}\mid{\frak p}}$) is the Frobenius element $\tau_{{\frak P}\mid{\frak p}}$. Since group actions correspond to conjugations on stabilizers, we have $D_{\sigma{\frak P}\mid{\frak p}}=\sigma D_{{\frak P}\mid{\frak p}}\sigma^{-1}$ for all $\sigma\in G_{L/K}$, and hence $\tau_{\sigma{\frak P}\mid{\frak p}}=\sigma\tau_{{\frak P}\mid{\frak p}}\sigma^{-1}$. As $\sigma$ varies over all of $G_{L/K}$, the primes $\sigma{\frak P}$ vary over all primes lying above $\frak p$, and $\tau$ traces out a full conjugacy class in $G_{L/K}$. This depends only on $\frak p$; call the class $F_{\frak p}$.

Let $S$ be the set of primes of $K$ unramified in $L$. We can put either the asymptotic or the Dirichlet density on $S$ - the first partially orders $S$ by norms and the second uses $L$-function residues. The second exists whenever the first does, but not necessarily vice-versa, and they are equal whenever both exist (hence Dirichlet density is easier to work with but a claim about asymptotic density is stronger than one about Dirichlet density). The Chebotarev Density theoerm asserts that the preimage of a conjugacy class $C\subseteq G_{L/K}$ under the map $F:S\to G_{L/K}$ has density $|C|/|G_{L/K}|$, using either density. There is another way to state this. Let $R$ be the primes in $L$ lying above those in $S$. Chebotarev asserts that if we push the densities of the fibers of $\tau:R\to G_{L/K}$ down to a probability distribution on $G_{L/K}$, we get the uniform distribution.

The splitting type of ${\frak p}\triangleleft{\frak O}_K$ in $L$ is the multiset of residue degrees $\{f_1,\cdots,f_g\}$. We can also speak of the isomorphism type of $\{{\frak P}_1,\cdots,{\frak P}_g\}$ as a $G_{L/K}$-set. The latter is actually a stronger invariant; two primes can have the same splitting type but the primes above them form different isoclasses of $G_{L/K}$-sets (this occurs when there are nonconjugate elements of $G_{L/K}$ of equal order). At first glance the density theorem seems incapable of saying anything about splitting types, or Galois isoclasses or about non-Galois extensions. However, it can be leveraged for all of these purposes.

Let $L/K$ be a separable extension and ${\frak p}$ an unramified prime of ${\frak O}_K$. Let $M$ be the Galois closure of this extension, so $M$ is Galois over both $L$ and $K$. Assume the following data:

• $[M:L]=m$
• ${\frak p}{\frak O}_L=\wp_1\cdots\wp_h$
• $\displaystyle\left[\frac{{\frak O}_L}{\wp_i}:\frac{{\frak O}_K}{{\frak p}}\right]=r_i$
• $\wp_i{\frak O}_M={\frak P}_{i,1}\cdots{\frak P}_{i,g_{\large i}}$
• $\displaystyle\left[\frac{{\frak O}_M}{{\frak P}_{i,j}}:\frac{{\frak O}_L}{\wp_i}\right]=f_i$
• $\displaystyle\left[\frac{{\frak O}_M}{{\frak P}_{i,j}}:\frac{{\frak O}_K}{{\frak p}}\right]=f$

We omit subscripts whenever it is irrelevant which prime we are talking about. In particular since we know $M$ is Galois over $L$ and $K$ we know $f_i$ doesn't depend on $j$ and $f$ on neither $i$ nor $j$.

This information may be summarized by the following diagram:

Keep in mind that all of this information stems simply from a single extension $L/K$ and a chosen prime ${\frak p}$ of ${\frak O}_K$. There are only finitely many primes of $K$ which do not ramify in $M$ (these are the prime ideals dividing the relative discriminant $\Delta_{M/K}\triangleleft{\frak O}_K$) so these can be safely ignored in any discussion on the densities of primes (we assume $K$ is a global field).

We know that Galois groups act transitively on primes-over-primes, so $\{{\frak P}_{i,1},\cdots,{\frak P}_{i,g_{\large i}}\}$ is one whole $G_{M/L}$-orbit for each $i$, and $\{\{{\frak P}_{1,1},\cdots,{\frak P}_{1,g_1}\},\cdots,\{{\frak P}_{h,1},\cdots,{\frak P}_{h,g_{\large h}}\}\}$ is the $G_{M/L}$-orbit partition of the set $S$ of all primes $\frak P$ of ${\frak O}_M$ lying above $\frak p$ in ${\frak O}_K$. Again, since $G_{M/K}$ acts transitively on $S$ we can say $S\cong G_{M/K}/D_{{\frak P}|{\frak p}}$ are isomorphic as $G_{M/K}$-sets.

We know that Galois groups act transitively on primes-over-primes, so $\{{\frak P}_{i,1},\cdots,{\frak P}_{i,g_{\large i}}\}$ is one whole $G_{M/L}$-orbit for each $i$, and $\{\{{\frak P}_{1,1},\cdots,{\frak P}_{1,g_1}\},\cdots,\{{\frak P}_{h,1},\cdots,{\frak P}_{h,g_{\large h}}\}\}$ is the $G_{M/L}$-orbit partition of the set $S$ of all primes $\frak P$ of ${\frak O}_M$ lying above $\frak p$ in ${\frak O}_K$. As $G_{M/K}$ is transitive on $S$ we can say $S\cong G_{M/K}/D_{{\frak P}|{\frak p}}$ are isomorphic as $G_{M/K}$-sets. Our data satisfy the relations $f_ig_i=m$ and $r_if_i=f$, hence $r_i=(f/m)g_i$ and $g_i=(m/f)r_i$. Therefore the splitting type of $\frak p$ in $L$ can be read off of the cell sizes of the $G_{M/K}$-orbit partition of $G_{L/K}/\langle\tau_{{\frak P}|\frak p}\rangle$ for any choice of prime ${\frak P}$ of $M$ lying above $\frak p$. But Chebotarev can say things about $\tau_{{\frak P}|{\frak p}}$! We may now conclude:

Extended Chebotarev Density Theorem. Say $M/K$ is Galois and $L$ intermediate. Let $H$ be a representative of a conjugacy class of subgroups of $G_{M/K}$. Then

1. The density of primes $\frak p$ in ${\frak O}_K$ for which the primes lying above them in ${\frak O}_M$ form a $G_{M/K}$ set with isomorphism type $G_{M/K}/H$ is the number of $\tau\in G_{M/K}$ for which $H$ is conjugate to $\langle\tau\rangle$ divided by $|G_{M/K}|$.
2. The density of primes $\frak p$ in ${\frak O}_K$ which have splitting type $\{r_1,\cdots,r_h\}$ in $L$ is equal to the number of $\tau\in G$ such that the cell size multiset of the $G_{M/L}$-orbit partition of $G_{M/K}/\langle\tau\rangle$ is given by $\{(m/|\tau|)r_1,\cdots,(m/|\tau|)r_h\}$ divided by $|G_{M/K}$.