Chebotarev density theorem

Let $L / K$ be a normal (finite-degree) extension of algebraic number fields with Galois group $\operatorname {Gal}( L / K )$. Pick a prime ideal $\frak P$ of $L$ and let $\mathfrak{p}$ be the prime ideal of $K$ under it, i.e. $\mathfrak { p } = A _ { K } \cap \mathfrak { P }$, where $A _ { K }$ is the ring of integers of $K$. There is a unique element

\begin{equation*} \sigma _ { \mathfrak { P } } = \left[ \frac { L / K } { \mathfrak { P } } \right] \end{equation*}

of $\operatorname {Gal}( L / K )$ such that $\sigma _ { \mathfrak { P } } \equiv x ^ { N ( \mathfrak { p } ) } \operatorname { mod } \mathfrak { P }$ for $x \in L$ integral. Here, $N ( \mathfrak{p} )$, the norm of $\mathfrak{p}$, is the number of elements of the residue field $A _ { K } / \mathfrak{p}$. This is the Frobenius automorphism (or Frobenius symbol) associated to $\frak P$.

If $\mathfrak{p}$ is unramified in $L / K$, define $F _ { L / K } ( \mathfrak{p} )$ as the conjugacy class of $\sigma _ { \mathfrak{P} }$ in $\operatorname {Gal}( L / K )$, where $\frak P$ is any prime ideal above $\mathfrak{p}$. This conjugacy class depends only on $\mathfrak{p}$.

The weak form of the Chebotarev density theorem says that if $A$ is an arbitrary conjugacy class in $\operatorname {Gal}( L / K )$, then the set

\begin{equation*} P _ { A } = \{ \mathfrak { p } : F _ { L / K} ( \mathfrak { p } ) = A \} \end{equation*}

is infinite and has Dirichlet density $\# A / n$, where $n = [ L : K ]$.

The stonger form specifies in addition that $P _ { A }$ is regular (see Dirichlet density) and that

\begin{equation*} N _ { A } = \left( \# \frac { A } { n } + o ( 1 ) \right) x \operatorname { log } x, \end{equation*}

with $N _ { A } ( x )$ the number of prime ideals in $P _ { A }$ with norm $\leq x$.

References

[a1]	W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.3 (Edition: Second)
[a2]	N.G. Chebotarev, "Determination of the density of the set of primes corresponding to a given class of permutations" Izv. Akad. Nauk. , 17 (1923) pp. 205–230; 231–250 (In Russian)
[a3]	N.G. Chebotarev, "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen welche zu einer gegebenen Substitutionsklasse gehören" Math. Ann. , 95 (1926) pp. 191–228