Difference between revisions of "Chebotarev density theorem"
From Encyclopedia of Mathematics
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| + | 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|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 $\mathfrak{p}$. | |
| − | is | + | 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|Dirichlet density]] $\# A / n$, where $n = [ L : K ]$. | |
| − | with | + | The stonger form specifies in addition that $P _ { A }$ is regular (see [[Dirichlet density|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==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.3 (Edition: Second)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> 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)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> N.G. Chebotarev, "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen welche zu einer gegebenen Substitutionsklasse gehören" ''Math. Ann.'' , '''95''' (1926) pp. 191–228</td></tr></table> |
Latest revision as of 17:00, 1 July 2020
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 |
How to Cite This Entry:
Chebotarev density theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebotarev_density_theorem&oldid=50363
Chebotarev density theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebotarev_density_theorem&oldid=50363
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article