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A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275701.png" /> obtained from the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275702.png" /> of rational numbers by adjoining a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275703.png" />-th root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275704.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275705.png" /> is a natural number. The term (local) cyclotomic field is also sometimes applied to the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275706.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275707.png" /> is the field of rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275708.png" />-adic numbers. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c0275709.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757010.png" /> is odd, it is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757012.png" />). Distinct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757013.png" /> then correspond to non-isomorphic fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757014.png" />.
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Cyclotomic fields arise naturally in the cyclotomy problem — the division of a circle into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757015.png" /> equal parts is equivalent to the construction of a primitive root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757016.png" /> in the complex plane. The structure of cyclotomic fields is "fairly simple" , and they therefore provide convenient experimental material in formulating general concepts in number theory. For example, the concept of an algebraic integer and a divisor first arose in the study of cyclotomic fields.
+
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 +
{{TEX|done}}
  
The special position of cyclotomic fields among all algebraic number fields is illustrated by the [[Kronecker–Weber theorem]], which states that a finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757017.png" /> is Abelian if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757018.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757019.png" />. An analogous proposition holds for local cyclotomic fields.
+
A field  $  K _ {n} = \mathbf Q ( \zeta _ {n} ) $
 +
obtained from the field  $  \mathbf Q $
 +
of rational numbers by adjoining a primitive  $  n $-th root of unity  $  \zeta _ {n} $,
 +
where  $  n $
 +
is a natural number. The term (local) cyclotomic field is also sometimes applied to the fields  $  \mathbf Q _ {p} ( \zeta _ {n} ) $,
 +
where  $  \mathbf Q _ {p} $
 +
is the field of rational  $  p $-adic numbers. Since  $  K _ {n} = K _ {2n} $
 +
when  $  n $
 +
is odd, it is usually assumed that  $  n \not\equiv 2 $ ($  \mathop{\rm mod}  4 $).
 +
Distinct  $  n $
 +
then correspond to non-isomorphic fields  $  K _ {n} $.
 +
 
 +
Cyclotomic fields arise naturally in the cyclotomy problem — the division of a circle into  $  n $
 +
equal parts is equivalent to the construction of a primitive root  $  \zeta _ {n} $
 +
in the complex plane. The structure of cyclotomic fields is "fairly simple" , and they therefore provide convenient experimental material in formulating general concepts in number theory. For example, the concept of an algebraic integer and a divisor first arose in the study of cyclotomic fields.
 +
 
 +
The special position of cyclotomic fields among all algebraic number fields is illustrated by the [[Kronecker–Weber theorem]], which states that a finite extension $  K/ \mathbf Q $
 +
is Abelian if and only if $  K \subset  K _ {n} $
 +
for some $  n $.  
 +
An analogous proposition holds for local cyclotomic fields.
  
 
==Algebraic theory.==
 
==Algebraic theory.==
The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757020.png" /> is an Abelian extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757021.png" /> with Galois group
+
The field $  K _ {n} $
 +
is an Abelian extension of $  \mathbf Q $
 +
with Galois group
  
<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/c027/c027570/c02757022.png" /></td> </tr></table>
+
$$
 +
G ( K _ {n} / \mathbf Q )  \simeq \
 +
( \mathbf Z /n \mathbf Z )  ^ {*} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757023.png" /> is the multiplicative group of the ring of residues modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757024.png" />. In particular, the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757025.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757027.png" /> is Euler's function. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757028.png" /> is totally imaginary and of degree 2 over its maximal totally real subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757029.png" />.
+
where $  ( \mathbf Z /n \mathbf Z )  ^ {*} $
 +
is the multiplicative group of the ring of residues modulo $  n $.  
 +
In particular, the degree $  [ K _ {n} : \mathbf Q ] $
 +
is $  \phi ( n) $,  
 +
where $  \phi ( n) $
 +
is Euler's function. The field $  K _ {n} $
 +
is totally imaginary and of degree 2 over its maximal totally real subfield $  K _ {n}  ^ {+} = \mathbf Q ( \zeta _ {n} + \zeta _ {n}  ^ {-1} ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757030.png" /> is the factorization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757031.png" /> into prime numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757032.png" /> is the linearly disjoint compositum of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757033.png" />. In the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757034.png" /> the prime divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757035.png" /> has ramification index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757036.png" />. In the same field one has the following equality of principal divisors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757037.png" />. All other prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757038.png" /> are unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757039.png" />, whence it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757040.png" /> is ramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757041.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757042.png" />.
+
If $  n = p _ {1} ^ {t _ {1} } \dots p _ {s} ^ {t _ {s} } $
 +
is the factorization of $  n $
 +
into prime numbers, then $  K _ {n} $
 +
is the linearly disjoint compositum of the fields $  K _ {p _ {1}  ^ {t _ {1} } } \dots K _ {p _ {s}  ^ {t _ {s} } } $.  
 +
In the field $  K _ {p  ^ {t}  } $
 +
the prime divisor $  p $
 +
has ramification index $  e = ( p - 1) p ^ {t - 1 } = \phi ( p  ^ {t} ) $.  
 +
In the same field one has the following equality of principal divisors: $  ( p) = ( 1 - \zeta _ {p  ^ {t}  } )  ^ {e} $.  
 +
All other prime divisors of $  \mathbf Q $
 +
are unramified in $  K _ {p  ^ {t}  } $,  
 +
whence it follows that $  p $
 +
is ramified in $  K _ {n} $
 +
if and only if $  p\mid  n $.
  
The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757043.png" /> form an integral basis for the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757044.png" />. The discriminant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757045.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757048.png" /> are fields which are linearly disjoint over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757049.png" /> with relatively prime discriminants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757054.png" />. This makes it possible to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757055.png" /> for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757056.png" /> (see [[#References|[3]]]).
+
The numbers $  1, \zeta _ {n} \dots \zeta _ {n} ^ {\phi ( n) - 1 } $
 +
form an integral basis for the field $  K _ {n} $.  
 +
The discriminant of $  K _ {p  ^ {t}  } $
 +
is equal to $  \pm  p ^ {p ^ {t - 1 } ( pt - t - 1) } $.  
 +
If $  E $
 +
and $  F $
 +
are fields which are linearly disjoint over $  \mathbf Q $
 +
with relatively prime discriminants $  D _ {E} $
 +
and $  D _ {F} $,  
 +
then $  D _ {EF} = D _ {E}  ^ {n} D _ {F}  ^ {m} $,  
 +
where $  n = [ F: \mathbf Q ] $,  
 +
$  m = [ E: \mathbf Q ] $.  
 +
This makes it possible to calculate $  D _ {K _ {n}  } $
 +
for arbitrary $  n $ (see [[#References|[3]]]).
  
For the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757057.png" />, the numbers
+
For the field $  K _ {p  ^ {t}  } $,  
 +
the numbers
  
<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/c027/c027570/c02757058.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757059.png" />, generate a subgroup of finite index in the group of all units. The elements of this subgroup are known as circular units or cyclotomic units.
+
\frac{( 1 - \zeta _ {p  ^ {t}  }  ^ {a} ) }{( 1 - \zeta _ {p  ^ {t}  }
 +
^ {b} ) }
 +
,
 +
$$
  
The decomposition law for cyclotomic fields, that is, the law according to which the prime divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757061.png" /> factorize into prime divisors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757062.png" />, is a particular case of the general decomposition law in Abelian extensions, established in class field theory (see [[#References|[4]]]). Explicitly: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757063.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757064.png" /> is the least natural number such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757065.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757066.png" />), then in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757067.png" />,
+
where  $  a, b \not\equiv 0 (  \mathop{\rm mod}  p) $,  
 +
generate a subgroup of finite index in the group of all units. The elements of this subgroup are known as circular units or cyclotomic units.
  
<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/c027/c027570/c02757068.png" /></td> </tr></table>
+
The decomposition law for cyclotomic fields, that is, the law according to which the prime divisors  $  ( p) $
 +
in  $  \mathbf Q $
 +
factorize into prime divisors in  $  K _ {n} $,
 +
is a particular case of the general decomposition law in Abelian extensions, established in class field theory (see [[#References|[4]]]). Explicitly: If  $  ( p, n) = 1 $
 +
and if  $  f $
 +
is the least natural number such that  $  p  ^ {f} \equiv 1 $ ($  \mathop{\rm mod}  n $),
 +
then in  $  K _ {n} $,
  
where the prime divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757069.png" /> are pairwise distinct, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757071.png" />. Thus, the factorization type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757072.png" /> depends only on the residue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757073.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757074.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757075.png" />, the exact form of the factorization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757076.png" /> can be obtained, using the facts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757079.png" /> is totally ramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757080.png" />.
+
$$
 +
( p) = \
 +
\mathfrak p _ {1} \dots \mathfrak p _ {g} ,
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757081.png" /> is the maximal Abelian extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757083.png" /> and
+
where the prime divisors  $  \mathfrak p _ {1} \dots \mathfrak p _ {g} $
 +
are pairwise distinct,  $  N ( \mathfrak p _ {i} ) = p  ^ {f} $
 +
and  $  fg = \phi ( n) $.  
 +
Thus, the factorization type of  $  ( p) $
 +
depends only on the residue of  $  p $ ($  \mathop{\rm mod}  n $).  
 +
If  $  p\mid  n $,
 +
the exact form of the factorization of $  ( p) $
 +
can be obtained, using the facts that  $  K _ {n} = K _ {m} K _ {p  ^ {t}  } $,
 +
where  $  ( m, p) = 1 $
 +
and  $  ( p) $
 +
is totally ramified in  $  K _ {p  ^ {t}  } $.
  
<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/c027/c027570/c02757084.png" /></td> </tr></table>
+
If  $  K $
 +
is the maximal Abelian extension of  $  \mathbf Q $,
 +
then  $  K = \cup _ {n} K _ {n} $
 +
and
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757085.png" /> is the completion of the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757086.png" /> with respect to all ideals of finite index. In particular, for any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757087.png" /> there is a unique extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757088.png" /> with Galois group isomorphic to the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757089.png" />-adic integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757090.png" />.
+
$$
 +
G ( K/ \mathbf Q )  \simeq \
 +
\lim\limits _  \leftarrow  G ( K _ {n} / \mathbf Q )  \simeq \
 +
\lim\limits  ( \mathbf Z /n \mathbf Z )  ^ {*}  \simeq \
 +
\widehat{\mathbf Z}  {}  ^ {*} ,
 +
$$
 +
 
 +
where  $  \widehat{\mathbf Z}  $
 +
is the completion of the ring of integers $  \mathbf Z $
 +
with respect to all ideals of finite index. In particular, for any prime number $  l $
 +
there is a unique extension $  \mathbf Q _ {\infty , l }  / \mathbf Q $
 +
with Galois group isomorphic to the group of $  l $-adic integers $  \mathbf Z _ {l} $.
  
 
According to [[Class field theory|class field theory]], there exists a reciprocity map
 
According to [[Class field theory|class field theory]], there exists a reciprocity map
  
<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/c027/c027570/c02757091.png" /></td> </tr></table>
+
$$
 +
\psi : J _ {\mathbf Q}  \rightarrow \
 +
G ( K/ \mathbf Q ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757092.png" /> is the idèle group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757093.png" />. In the case of a cyclotomic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757094.png" /> admits a simple explicit description (see [[#References|[4]]]).
+
where $  J _ {\mathbf Q} $
 +
is the idèle group of $  \mathbf Q $.  
 +
In the case of a cyclotomic field $  \psi $
 +
admits a simple explicit description (see [[#References|[4]]]).
  
 
==Analytic theory.==
 
==Analytic theory.==
Many results regarding the structure of the divisor class group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757095.png" /> can be proved by analytic methods. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757096.png" /> is the class number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757097.png" />, then
+
Many results regarding the structure of the divisor class group of $  K _ {n} $
 +
can be proved by analytic methods. If $  h _ {n} $
 +
is the class number of $  K _ {n} $,  
 +
then
  
<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/c027/c027570/c02757098.png" /></td> </tr></table>
+
$$
 +
h _ {n}  = \
 +
2  ^ {-t} \pi  ^ {-t} R  ^ {-1} w
 +
\sqrt {| D | }
 +
\prod _ {\chi \neq \chi _ {0} }
 +
L ( 1, \chi ),
 +
$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c02757099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570100.png" /> are, respectively, the number of roots of unity, the discriminant and the regulator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570102.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570103.png" /> is the Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570104.png" />-function for the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570106.png" /> runs through all non-trivial primitive multiplicative characters modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570107.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570108.png" /> in turn can be expressed explicitly in terms of Gauss sums (see [[#References|[7]]]). This solves the problem of calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570109.png" />, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570110.png" />.
+
here $  w, D $
 +
and $  R $
 +
are, respectively, the number of roots of unity, the discriminant and the regulator of $  K _ {n} $,  
 +
$  t = \phi ( n)/2 $,  
 +
and $  L ( 1, \chi ) $
 +
is the Dirichlet $  L $-function for the character $  \chi $,  
 +
where $  \chi $
 +
runs through all non-trivial primitive multiplicative characters modulo $  n $.  
 +
The function $  L ( 1, \chi ) $
 +
in turn can be expressed explicitly in terms of Gauss sums (see [[#References|[7]]]). This solves the problem of calculating $  h _ {n} $,  
 +
given $  n $.
  
There is a natural decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570111.png" /> into two factors, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570112.png" />; the first and second factor of the class number, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570113.png" /> is interpreted as the class number of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570114.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570115.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570116.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570117.png" /> is the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570119.png" /> is the group of real cyclotomic units (any cyclotomic unit becomes real if multiplied by a suitable root of unity).
+
There is a natural decomposition of $  h _ {n} $
 +
into two factors, $  h _ {n} = h _ {n}  ^ {-} h _ {n}  ^ {+} $;  
 +
the first and second factor of the class number, where $  h _ {n}  ^ {+} $
 +
is interpreted as the class number of the field $  K _ {n}  ^ {+} $.  
 +
If $  n = p  ^ {t} $
 +
then  $  h _ {n}  ^ {+} = [ E : E _ {0} ] $,  
 +
where $  E $
 +
is the group of units of $  K _ {p  ^ {t}  }  ^ {+} $
 +
and $  E _ {0} $
 +
is the group of real cyclotomic units (any cyclotomic unit becomes real if multiplied by a suitable root of unity).
  
In questions related to the Fermat problem, an important role is played by the divisibility of the class number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570120.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570121.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570122.png" /> is prime. It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570123.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570124.png" />) for infinitely many prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570125.png" /> (such numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570126.png" /> are said to be irregular). As to the set of regular prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570127.png" />, i.e. numbers for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570128.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570129.png" />), it is not known (1982) whether it is finite or infinite. It has been conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570130.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570131.png" />) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570132.png" />, and this has been confirmed in a large number of cases. The factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570133.png" /> is more amenable to investigation. There exists a relatively simple criterion for the divisibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570134.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570135.png" />) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570136.png" /> in terms of Bernoulli numbers ([[#References|[7]]]). It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570137.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570138.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570139.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570140.png" /> (see [[#References|[6]]]).
+
In questions related to the Fermat problem, an important role is played by the divisibility of the class number of $  K _ {l} $
 +
by $  l $,  
 +
where $  l $
 +
is prime. It is known that $  h _ {l} \equiv 0 $ ($  \mathop{\rm mod}  l $)  
 +
for infinitely many prime numbers $  l $ (such numbers $  l $
 +
are said to be irregular). As to the set of regular prime numbers $  l $,  
 +
i.e. numbers for which $  h _ {l} \not\equiv 0 $ ($  \mathop{\rm mod}  l $),  
 +
it is not known (1982) whether it is finite or infinite. It has been conjectured that $  h _ {l}  ^ {+} \not\equiv 0 $ ($  \mathop{\rm mod}  l $)  
 +
for all $  l $,  
 +
and this has been confirmed in a large number of cases. The factor $  h _ {l}  ^ {-} $
 +
is more amenable to investigation. There exists a relatively simple criterion for the divisibility of $  h _ {l}  ^ {-} $ (and $  h _ {l} $)  
 +
by $  l $
 +
in terms of Bernoulli numbers ([[#References|[7]]]). It is known that $  h _ {l}  ^ {-} \rightarrow \infty $
 +
as $  l \rightarrow \infty $
 +
and that $  h _ {l} = 1 $
 +
if and only if $  l \leq  19 $ (see [[#References|[6]]]).
  
The so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570141.png" />-adic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570142.png" />-functions have been successfully applied to the study of the class groups of cyclotomic fields (see [[#References|[5]]], [[#References|[8]]]).
+
The so-called $  p $-adic $  L $-functions have been successfully applied to the study of the class groups of cyclotomic fields (see [[#References|[5]]], [[#References|[8]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Kummer, "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten komplexen Zahlen in ihre Primfaktoren" ''J. Reine Angew. Math.'' , '''35''' (1847) pp. 327–367</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) {{MR|1617068}} {{MR|0002354}} {{ZBL|0921.11001}} {{ZBL|0063.08223}} {{ZBL|0054.02002}} {{ZBL|66.1210.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Algebraic number theory" , Addison-Wesley (1970) {{MR|0282947}} {{ZBL|0211.38404}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967) {{MR|0215665}} {{ZBL|0153.07403}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> K. Uchida, "Class numbers of imaginary abelian number fields III" ''Tôhoku Math. J.'' , '''23''' (1971) pp. 573–580 {{MR|0288097}} {{ZBL|0241.12002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|1355542}} {{MR|1534414}} {{MR|0195803}} {{ZBL|0614.00005}} {{ZBL|0592.12001}} {{ZBL|0145.04901}} {{ZBL|0145.04902}} {{ZBL|0121.04202}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> K. Iwasawa, "Lectures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570143.png" />-adic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570144.png" />-functions" , Springer (1972) {{MR|360526}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. Lang, "Cyclotomic fields" , Springer (1978) {{MR|0485768}} {{ZBL|0395.12005}} </TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Kummer, "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten komplexen Zahlen in ihre Primfaktoren" ''J. Reine Angew. Math.'' , '''35''' (1847) pp. 327–367</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) {{MR|1617068}} {{MR|0002354}} {{ZBL|0921.11001}} {{ZBL|0063.08223}} {{ZBL|0054.02002}} {{ZBL|66.1210.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Algebraic number theory" , Addison-Wesley (1970) {{MR|0282947}} {{ZBL|0211.38404}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967) {{MR|0215665}} {{ZBL|0153.07403}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> K. Uchida, "Class numbers of imaginary abelian number fields III" ''Tôhoku Math. J.'' , '''23''' (1971) pp. 573–580 {{MR|0288097}} {{ZBL|0241.12002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|1355542}} {{MR|1534414}} {{MR|0195803}} {{ZBL|0614.00005}} {{ZBL|0592.12001}} {{ZBL|0145.04901}} {{ZBL|0145.04902}} {{ZBL|0121.04202}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> K. Iwasawa, "Lectures on $p$-adic L-functions" , Springer (1972) {{MR|360526}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. Lang, "Cyclotomic fields" , Springer (1978) {{MR|0485768}} {{ZBL|0395.12005}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
It is still not known whether there are infinitely many regular primes (1987), but it is conjectured that the density of the set of regular primes inside the set of all primes is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570145.png" />, see [[#References|[a5]]]. It has been proved (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570146.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570147.png" />) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570148.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570149.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570150.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570151.png" />, see [[#References|[a3]]], [[#References|[a5]]]. An important theorem that describes the structure of the class groups of cyclotomic fields in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570152.png" />-adic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570153.png" />-functions has been proved by B. Mazur and A. Wiles (see [[#References|[a1]]], [[#References|[a4]]]) with the help of methods from algebraic geometry.
+
It is still not known whether there are infinitely many regular primes (1987), but it is conjectured that the density of the set of regular primes inside the set of all primes is $  e  ^ {-1/2} \approx 0.6065 $,  
 +
see [[#References|[a5]]]. It has been proved (for $  n \not\equiv 2 $
 +
$  \mathop{\rm mod}  4 $)  
 +
that $  h _ {n} = 1 $
 +
if and only if $  \phi ( n) \leq  20 $
 +
or $  n = 35, 45 $
 +
or $  84 $,  
 +
see [[#References|[a3]]], [[#References|[a5]]]. An important theorem that describes the structure of the class groups of cyclotomic fields in terms of $  p $-adic $  L $-functions has been proved by B. Mazur and A. Wiles (see [[#References|[a1]]], [[#References|[a4]]]) with the help of methods from algebraic geometry.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Coates, "The work of Mazur and Wiles on cyclotomic fields." ''Sem. Bourbaki'' , '''33''' : 575 (1980/81) ''Lecture Notes in Math.'' , '''901''' (1981) pp. 220–242 {{MR|0647499}} {{ZBL|0506.12001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang, "Cyclotomic fields" , '''II''' , Springer (1980) {{MR|0566952}} {{ZBL|0435.12001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.M. Masley, H.L. Montgomery, "Cyclotomic fields with unique factorization" ''J. Reine Angew. Math.'' , '''286–287''' (1976) pp. 248–256 {{MR|0429824}} {{ZBL|0335.12013}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> B. Mazur, A. Wiles, "Class fields of abelian extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027570/c027570154.png" />" ''Invent. Math.'' , '''76''' (1984) pp. 179–330 {{MR|0742853}} {{ZBL|0545.12005}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L.C. Washington, "Cyclotomic fields" , Springer (1982) {{MR|0718674}} {{ZBL|0502.12003}} {{ZBL|0484.12001}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Coates, "The work of Mazur and Wiles on cyclotomic fields." ''Sem. Bourbaki'' , '''33''' : 575 (1980/81) ''Lecture Notes in Math.'' , '''901''' (1981) pp. 220–242 {{MR|0647499}} {{ZBL|0506.12001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang, "Cyclotomic fields" , '''II''' , Springer (1980) {{MR|0566952}} {{ZBL|0435.12001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.M. Masley, H.L. Montgomery, "Cyclotomic fields with unique factorization" ''J. Reine Angew. Math.'' , '''286–287''' (1976) pp. 248–256 {{MR|0429824}} {{ZBL|0335.12013}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> B. Mazur, A. Wiles, "Class fields of abelian extensions of $\QQ$" ''Invent. Math.'' , '''76''' (1984) pp. 179–330 {{MR|0742853}} {{ZBL|0545.12005}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L.C. Washington, "Cyclotomic fields" , Springer (1982) {{MR|0718674}} {{ZBL|0502.12003}} {{ZBL|0484.12001}} </TD></TR></table>

Latest revision as of 11:19, 26 March 2023


A field $ K _ {n} = \mathbf Q ( \zeta _ {n} ) $ obtained from the field $ \mathbf Q $ of rational numbers by adjoining a primitive $ n $-th root of unity $ \zeta _ {n} $, where $ n $ is a natural number. The term (local) cyclotomic field is also sometimes applied to the fields $ \mathbf Q _ {p} ( \zeta _ {n} ) $, where $ \mathbf Q _ {p} $ is the field of rational $ p $-adic numbers. Since $ K _ {n} = K _ {2n} $ when $ n $ is odd, it is usually assumed that $ n \not\equiv 2 $ ($ \mathop{\rm mod} 4 $). Distinct $ n $ then correspond to non-isomorphic fields $ K _ {n} $.

Cyclotomic fields arise naturally in the cyclotomy problem — the division of a circle into $ n $ equal parts is equivalent to the construction of a primitive root $ \zeta _ {n} $ in the complex plane. The structure of cyclotomic fields is "fairly simple" , and they therefore provide convenient experimental material in formulating general concepts in number theory. For example, the concept of an algebraic integer and a divisor first arose in the study of cyclotomic fields.

The special position of cyclotomic fields among all algebraic number fields is illustrated by the Kronecker–Weber theorem, which states that a finite extension $ K/ \mathbf Q $ is Abelian if and only if $ K \subset K _ {n} $ for some $ n $. An analogous proposition holds for local cyclotomic fields.

Algebraic theory.

The field $ K _ {n} $ is an Abelian extension of $ \mathbf Q $ with Galois group

$$ G ( K _ {n} / \mathbf Q ) \simeq \ ( \mathbf Z /n \mathbf Z ) ^ {*} , $$

where $ ( \mathbf Z /n \mathbf Z ) ^ {*} $ is the multiplicative group of the ring of residues modulo $ n $. In particular, the degree $ [ K _ {n} : \mathbf Q ] $ is $ \phi ( n) $, where $ \phi ( n) $ is Euler's function. The field $ K _ {n} $ is totally imaginary and of degree 2 over its maximal totally real subfield $ K _ {n} ^ {+} = \mathbf Q ( \zeta _ {n} + \zeta _ {n} ^ {-1} ) $.

If $ n = p _ {1} ^ {t _ {1} } \dots p _ {s} ^ {t _ {s} } $ is the factorization of $ n $ into prime numbers, then $ K _ {n} $ is the linearly disjoint compositum of the fields $ K _ {p _ {1} ^ {t _ {1} } } \dots K _ {p _ {s} ^ {t _ {s} } } $. In the field $ K _ {p ^ {t} } $ the prime divisor $ p $ has ramification index $ e = ( p - 1) p ^ {t - 1 } = \phi ( p ^ {t} ) $. In the same field one has the following equality of principal divisors: $ ( p) = ( 1 - \zeta _ {p ^ {t} } ) ^ {e} $. All other prime divisors of $ \mathbf Q $ are unramified in $ K _ {p ^ {t} } $, whence it follows that $ p $ is ramified in $ K _ {n} $ if and only if $ p\mid n $.

The numbers $ 1, \zeta _ {n} \dots \zeta _ {n} ^ {\phi ( n) - 1 } $ form an integral basis for the field $ K _ {n} $. The discriminant of $ K _ {p ^ {t} } $ is equal to $ \pm p ^ {p ^ {t - 1 } ( pt - t - 1) } $. If $ E $ and $ F $ are fields which are linearly disjoint over $ \mathbf Q $ with relatively prime discriminants $ D _ {E} $ and $ D _ {F} $, then $ D _ {EF} = D _ {E} ^ {n} D _ {F} ^ {m} $, where $ n = [ F: \mathbf Q ] $, $ m = [ E: \mathbf Q ] $. This makes it possible to calculate $ D _ {K _ {n} } $ for arbitrary $ n $ (see [3]).

For the field $ K _ {p ^ {t} } $, the numbers

$$ \frac{( 1 - \zeta _ {p ^ {t} } ^ {a} ) }{( 1 - \zeta _ {p ^ {t} } ^ {b} ) } , $$

where $ a, b \not\equiv 0 ( \mathop{\rm mod} p) $, generate a subgroup of finite index in the group of all units. The elements of this subgroup are known as circular units or cyclotomic units.

The decomposition law for cyclotomic fields, that is, the law according to which the prime divisors $ ( p) $ in $ \mathbf Q $ factorize into prime divisors in $ K _ {n} $, is a particular case of the general decomposition law in Abelian extensions, established in class field theory (see [4]). Explicitly: If $ ( p, n) = 1 $ and if $ f $ is the least natural number such that $ p ^ {f} \equiv 1 $ ($ \mathop{\rm mod} n $), then in $ K _ {n} $,

$$ ( p) = \ \mathfrak p _ {1} \dots \mathfrak p _ {g} , $$

where the prime divisors $ \mathfrak p _ {1} \dots \mathfrak p _ {g} $ are pairwise distinct, $ N ( \mathfrak p _ {i} ) = p ^ {f} $ and $ fg = \phi ( n) $. Thus, the factorization type of $ ( p) $ depends only on the residue of $ p $ ($ \mathop{\rm mod} n $). If $ p\mid n $, the exact form of the factorization of $ ( p) $ can be obtained, using the facts that $ K _ {n} = K _ {m} K _ {p ^ {t} } $, where $ ( m, p) = 1 $ and $ ( p) $ is totally ramified in $ K _ {p ^ {t} } $.

If $ K $ is the maximal Abelian extension of $ \mathbf Q $, then $ K = \cup _ {n} K _ {n} $ and

$$ G ( K/ \mathbf Q ) \simeq \ \lim\limits _ \leftarrow G ( K _ {n} / \mathbf Q ) \simeq \ \lim\limits ( \mathbf Z /n \mathbf Z ) ^ {*} \simeq \ \widehat{\mathbf Z} {} ^ {*} , $$

where $ \widehat{\mathbf Z} $ is the completion of the ring of integers $ \mathbf Z $ with respect to all ideals of finite index. In particular, for any prime number $ l $ there is a unique extension $ \mathbf Q _ {\infty , l } / \mathbf Q $ with Galois group isomorphic to the group of $ l $-adic integers $ \mathbf Z _ {l} $.

According to class field theory, there exists a reciprocity map

$$ \psi : J _ {\mathbf Q} \rightarrow \ G ( K/ \mathbf Q ), $$

where $ J _ {\mathbf Q} $ is the idèle group of $ \mathbf Q $. In the case of a cyclotomic field $ \psi $ admits a simple explicit description (see [4]).

Analytic theory.

Many results regarding the structure of the divisor class group of $ K _ {n} $ can be proved by analytic methods. If $ h _ {n} $ is the class number of $ K _ {n} $, then

$$ h _ {n} = \ 2 ^ {-t} \pi ^ {-t} R ^ {-1} w \sqrt {| D | } \prod _ {\chi \neq \chi _ {0} } L ( 1, \chi ), $$

here $ w, D $ and $ R $ are, respectively, the number of roots of unity, the discriminant and the regulator of $ K _ {n} $, $ t = \phi ( n)/2 $, and $ L ( 1, \chi ) $ is the Dirichlet $ L $-function for the character $ \chi $, where $ \chi $ runs through all non-trivial primitive multiplicative characters modulo $ n $. The function $ L ( 1, \chi ) $ in turn can be expressed explicitly in terms of Gauss sums (see [7]). This solves the problem of calculating $ h _ {n} $, given $ n $.

There is a natural decomposition of $ h _ {n} $ into two factors, $ h _ {n} = h _ {n} ^ {-} h _ {n} ^ {+} $; the first and second factor of the class number, where $ h _ {n} ^ {+} $ is interpreted as the class number of the field $ K _ {n} ^ {+} $. If $ n = p ^ {t} $ then $ h _ {n} ^ {+} = [ E : E _ {0} ] $, where $ E $ is the group of units of $ K _ {p ^ {t} } ^ {+} $ and $ E _ {0} $ is the group of real cyclotomic units (any cyclotomic unit becomes real if multiplied by a suitable root of unity).

In questions related to the Fermat problem, an important role is played by the divisibility of the class number of $ K _ {l} $ by $ l $, where $ l $ is prime. It is known that $ h _ {l} \equiv 0 $ ($ \mathop{\rm mod} l $) for infinitely many prime numbers $ l $ (such numbers $ l $ are said to be irregular). As to the set of regular prime numbers $ l $, i.e. numbers for which $ h _ {l} \not\equiv 0 $ ($ \mathop{\rm mod} l $), it is not known (1982) whether it is finite or infinite. It has been conjectured that $ h _ {l} ^ {+} \not\equiv 0 $ ($ \mathop{\rm mod} l $) for all $ l $, and this has been confirmed in a large number of cases. The factor $ h _ {l} ^ {-} $ is more amenable to investigation. There exists a relatively simple criterion for the divisibility of $ h _ {l} ^ {-} $ (and $ h _ {l} $) by $ l $ in terms of Bernoulli numbers ([7]). It is known that $ h _ {l} ^ {-} \rightarrow \infty $ as $ l \rightarrow \infty $ and that $ h _ {l} = 1 $ if and only if $ l \leq 19 $ (see [6]).

The so-called $ p $-adic $ L $-functions have been successfully applied to the study of the class groups of cyclotomic fields (see [5], [8]).

References

[1] E. Kummer, "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten komplexen Zahlen in ihre Primfaktoren" J. Reine Angew. Math. , 35 (1847) pp. 327–367
[2] H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) MR1617068 MR0002354 Zbl 0921.11001 Zbl 0063.08223 Zbl 0054.02002 Zbl 66.1210.02
[3] S. Lang, "Algebraic number theory" , Addison-Wesley (1970) MR0282947 Zbl 0211.38404
[4] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) MR0215665 Zbl 0153.07403
[5] I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)
[6] K. Uchida, "Class numbers of imaginary abelian number fields III" Tôhoku Math. J. , 23 (1971) pp. 573–580 MR0288097 Zbl 0241.12002
[7] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966) MR1355542 MR1534414 MR0195803 Zbl 0614.00005 Zbl 0592.12001 Zbl 0145.04901 Zbl 0145.04902 Zbl 0121.04202
[8] K. Iwasawa, "Lectures on $p$-adic L-functions" , Springer (1972) MR360526
[9] S. Lang, "Cyclotomic fields" , Springer (1978) MR0485768 Zbl 0395.12005

Comments

It is still not known whether there are infinitely many regular primes (1987), but it is conjectured that the density of the set of regular primes inside the set of all primes is $ e ^ {-1/2} \approx 0.6065 $, see [a5]. It has been proved (for $ n \not\equiv 2 $ $ \mathop{\rm mod} 4 $) that $ h _ {n} = 1 $ if and only if $ \phi ( n) \leq 20 $ or $ n = 35, 45 $ or $ 84 $, see [a3], [a5]. An important theorem that describes the structure of the class groups of cyclotomic fields in terms of $ p $-adic $ L $-functions has been proved by B. Mazur and A. Wiles (see [a1], [a4]) with the help of methods from algebraic geometry.

References

[a1] J. Coates, "The work of Mazur and Wiles on cyclotomic fields." Sem. Bourbaki , 33 : 575 (1980/81) Lecture Notes in Math. , 901 (1981) pp. 220–242 MR0647499 Zbl 0506.12001
[a2] S. Lang, "Cyclotomic fields" , II , Springer (1980) MR0566952 Zbl 0435.12001
[a3] J.M. Masley, H.L. Montgomery, "Cyclotomic fields with unique factorization" J. Reine Angew. Math. , 286–287 (1976) pp. 248–256 MR0429824 Zbl 0335.12013
[a4] B. Mazur, A. Wiles, "Class fields of abelian extensions of $\QQ$" Invent. Math. , 76 (1984) pp. 179–330 MR0742853 Zbl 0545.12005
[a5] L.C. Washington, "Cyclotomic fields" , Springer (1982) MR0718674 Zbl 0502.12003 Zbl 0484.12001
How to Cite This Entry:
Cyclotomic field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclotomic_field&oldid=39901
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article