We say that

is an

__abelian extension__ iff it is a Galois extension with

an abelian group. Now if

, where

is a field, then

is a normal subgroup of

therefore by the fundamental theorem of Galois theory it follows that

must be a normal extension.

Cyclotomic extensions are a special case of abelian extensions. Thus for

to be a subfield of a cyclotomic field we require for

to be a Galois extension with an abelian Galois group. This is obviously not true because the minimal polynomial of

is

, and only one root of the three is contained in

.

By the way there is an incredible theorem in algebraic number theory. Above we said that for

to be contained in a cyclotomic field we require for

to be an (finite) abelian extension. However, the converse is true!!! In other words, if

is a (finite) abelian extension then it is a subfield of some cyclotomic extension. This deep result is known as the Kronecker-Weber theorem.