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''of a field  $  k $''

An extension  $  K $
obtained from  $  k $
by adjunction of a root of unity (cf. [[Primitive root|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|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 [[#References|[2]]], [[#References|[3]]], [[#References|[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====
<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>