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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" />.
 
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" />.
+
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
 
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

Revision as of 20:57, 29 November 2014

Let be a normal (finite-degree) extension of algebraic number fields with Galois group . Pick a prime ideal of and let be the prime ideal of under it, i.e. , where is the ring of integers of . There is a unique element

of such that for integral. Here, , the norm of , is the number of elements of the residue field . This is the Frobenius automorphism (or Frobenius symbol) associated to .

If is unramified in , define as the conjugacy class of in , where is any prime ideal above . This conjugacy class depends only on .

The weak form of the Chebotarev density theorem says that if is an arbitrary conjugacy class in , then the set

is infinite and has Dirichlet density , where .

The stonger form specifies in addition that is regular (see Dirichlet density) and that

with the number of prime ideals in with norm .

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=35122
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article