# 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
How to Cite This Entry:
Cyclotomic extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclotomic_extension&oldid=51180
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