Cyclotomic extension

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