Cyclotomic extension

of a field

An extension obtained from by adjunction of a root of unity (cf. Primitive root). The term is sometimes used for any subextension of over . 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 is the field of rational numbers.

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

If is an algebraic number field, the only prime divisors that may be ramified in are those dividing , although when a divisor of dividing may remain unramified in . The cyclotomic extension of an algebraic number field with Galois group isomorphic to the additive group of -adic numbers is called the cyclotomic -extension (see [2], [3], [4]). In the case this -extension has the form , where .

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 -extensions of algebraic number fields" Ann. of Math. , 98 : 2 (1973) pp. 246–326