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.