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''of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275601.png" />''
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An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275602.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275603.png" /> by adjunction of a root of unity (cf. [[Primitive root|Primitive root]]). The term is sometimes used for any subextension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275604.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275605.png" />. An infinite algebraic extension which is the union of finite cyclotomic extensions is also called a cyclotomic extension. Important examples of cyclotomic extensions are provided by the cyclotomic fields (cf. [[Cyclotomic field|Cyclotomic field]]), obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275606.png" /> is the field of rational numbers.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275607.png" /> be of characteristic 0 and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275608.png" /> be a cyclotomic extension obtained by adjoining a primitive root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c0275609.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756010.png" /> is the composite of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756011.png" /> and the cyclotomic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756012.png" />. For this reason, many properties of cyclotomic fields carry over to cyclotomic extensions. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756013.png" /> is an Abelian extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756014.png" /> (this is also true for fields of finite characteristic), the Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756015.png" /> is a subgroup of the Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756016.png" />; in particular, the order of the former Galois group divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756018.png" /> is the Euler function.
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''of a field $  k $''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756019.png" /> is an algebraic number field, the only prime divisors that may be ramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756020.png" /> are those dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756021.png" />, although when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756022.png" /> a divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756023.png" /> dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756024.png" /> may remain unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756025.png" />. The cyclotomic extension of an algebraic number field with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756026.png" /> isomorphic to the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756028.png" />-adic numbers is called the cyclotomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756030.png" />-extension (see [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]). In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756031.png" /> this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756032.png" />-extension has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756034.png" />.
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An extension  $  K $
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obtained from  $  k $
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by adjunction of a root of unity (cf. [[Primitive root|Primitive root]]). The term is sometimes used for any subextension of  $  K $
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over  $  k $.
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An infinite algebraic extension which is the union of finite cyclotomic extensions is also called a cyclotomic extension. Important examples of cyclotomic extensions are provided by the cyclotomic fields (cf. [[Cyclotomic field|Cyclotomic field]]), obtained when  $  k = \mathbf Q $
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is the field of rational numbers.
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Let  $  k $
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be of characteristic 0 and let  $  k ( \zeta _ {n} ) $
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be a cyclotomic extension obtained by adjoining a primitive root of unity  $  \zeta _ {n} $.
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Then  $  k ( \zeta _ {n} ) $
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is the composite of  $  k $
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and the cyclotomic field  $  \mathbf Q ( \zeta _ {n} ) $.  
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For this reason, many properties of cyclotomic fields carry over to cyclotomic extensions. For example,  $  k ( \zeta _ {n} ) $
 +
is an Abelian extension of  $  k $(
 +
this is also true for fields of finite characteristic), the Galois group of  $  k ( \zeta _ {n} )/k $
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is a subgroup of the Galois group of  $  \mathbf Q ( \zeta _ {n} )/ \mathbf Q $;
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in particular, the order of the former Galois group divides  $  \phi ( n) $,
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where  $  \phi ( n) $
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is the Euler function.
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If  $  k $
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is an algebraic number field, the only prime divisors that may be ramified in $  k ( \zeta _ {n} )/k $
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are those dividing $  n $,  
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although when $  k \neq \mathbf Q $
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a divisor of $  k $
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dividing $  n $
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may remain unramified in $  k ( \zeta _ {n} ) $.  
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The cyclotomic extension of an algebraic number field with Galois group $  \Gamma $
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isomorphic to the additive group $  \mathbf Z _ {l} $
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of $  l $-
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adic numbers is called the cyclotomic $  \Gamma $-
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extension (see [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]). In the case $  \zeta _ {l} \in k $
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this $  \Gamma $-
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extension has the form $  k _  \infty  = \cup _ {n} k _ {n} $,  
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where $  k _ {n} = k ( \zeta _ {l  ^ {n}  } ) $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,  "The zeta-function" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kuz'min,  "The Tate module of algebraic number fields"  ''Izv. Akad. Nauk SSSR'' , '''36''' :  2  (1972)  pp. 267–327  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Iwasawa,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027560/c02756035.png" />-extensions of algebraic number fields"  ''Ann. of Math.'' , '''98''' :  2  (1973)  pp. 246–326</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,  "The zeta-function" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kuz'min,  "The Tate module of algebraic number fields"  ''Izv. Akad. Nauk SSSR'' , '''36''' :  2  (1972)  pp. 267–327  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Iwasawa,  "On $\mathbf Z_{l}$-extensions of algebraic number fields"  ''Ann. of Math.'' , '''98''' :  2  (1973)  pp. 246–326</TD></TR></table>

Latest revision as of 18:23, 2 January 2021


of a field $ k $

An extension $ K $ obtained from $ k $ by adjunction of a root of unity (cf. Primitive root). The term is sometimes used for any subextension of $ K $ over $ k $. An infinite algebraic extension which is the union of finite cyclotomic extensions is also called a cyclotomic extension. Important examples of cyclotomic extensions are provided by the cyclotomic fields (cf. Cyclotomic field), obtained when $ k = \mathbf Q $ is the field of rational numbers.

Let $ k $ be of characteristic 0 and let $ k ( \zeta _ {n} ) $ be a cyclotomic extension obtained by adjoining a primitive root of unity $ \zeta _ {n} $. Then $ k ( \zeta _ {n} ) $ is the composite of $ k $ and the cyclotomic field $ \mathbf Q ( \zeta _ {n} ) $. For this reason, many properties of cyclotomic fields carry over to cyclotomic extensions. For example, $ k ( \zeta _ {n} ) $ is an Abelian extension of $ k $( this is also true for fields of finite characteristic), the Galois group of $ k ( \zeta _ {n} )/k $ is a subgroup of the Galois group of $ \mathbf Q ( \zeta _ {n} )/ \mathbf Q $; in particular, the order of the former Galois group divides $ \phi ( n) $, where $ \phi ( n) $ is the Euler function.

If $ k $ is an algebraic number field, the only prime divisors that may be ramified in $ k ( \zeta _ {n} )/k $ are those dividing $ n $, although when $ k \neq \mathbf Q $ a divisor of $ k $ dividing $ n $ may remain unramified in $ k ( \zeta _ {n} ) $. The cyclotomic extension of an algebraic number field with Galois group $ \Gamma $ isomorphic to the additive group $ \mathbf Z _ {l} $ of $ l $- adic numbers is called the cyclotomic $ \Gamma $- extension (see [2], [3], [4]). In the case $ \zeta _ {l} \in k $ this $ \Gamma $- extension has the form $ k _ \infty = \cup _ {n} k _ {n} $, where $ k _ {n} = k ( \zeta _ {l ^ {n} } ) $.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974)
[2] I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)
[3] L.V. Kuz'min, "The Tate module of algebraic number fields" Izv. Akad. Nauk SSSR , 36 : 2 (1972) pp. 267–327 (In Russian)
[4] K. Iwasawa, "On $\mathbf Z_{l}$-extensions of algebraic number fields" Ann. of Math. , 98 : 2 (1973) pp. 246–326
How to Cite This Entry:
Cyclotomic extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclotomic_extension&oldid=14022
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