View source for Chebotarev density theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300801.png" /> be a normal (finite-degree) extension of algebraic number fields with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300802.png" />. Pick a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300803.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300804.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300805.png" /> be the prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300806.png" /> under it, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300807.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300808.png" /> is the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300809.png" />. There is a unique element

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008010.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008013.png" /> integral. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008014.png" />, the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008015.png" />, is the number of elements of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008016.png" />. This is the [[Frobenius automorphism|Frobenius automorphism]] (or Frobenius symbol) associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008017.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008018.png" /> is unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008019.png" />, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008020.png" /> as the conjugacy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008023.png" /> is any prime ideal above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008024.png" />. This conjugacy class depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008025.png" />.

The weak form of the Chebotarev density theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008026.png" /> is an arbitrary conjugacy class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008027.png" />, then the set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008028.png" /></td> </tr></table>

is infinite and has [[Dirichlet density|Dirichlet density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008030.png" />.

The stonger form specifies in addition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008031.png" /> is regular (see [[Dirichlet density|Dirichlet density]]) and that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008032.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008033.png" /> the number of prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008034.png" /> with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008035.png" />.

====References====
<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>