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**ThePerfectHacker** We say that $\displaystyle K/\mathbb{Q}$ is an __abelian extension__ iff it is a Galois extension with $\displaystyle \text{Gal}(K/\mathbb{Q})$ an abelian group. Now if $\displaystyle \mathbb{Q}\subseteq L \subseteq K$, where $\displaystyle L$ is a field, then $\displaystyle \text{Gal}(K/L)$ is a normal subgroup of $\displaystyle \text{Gal}(K/\mathbb{Q})$ therefore by the fundamental theorem of Galois theory it follows that $\displaystyle L/\mathbb{Q}$ must be a normal extension.

Cyclotomic extensions are a special case of abelian extensions. Thus for $\displaystyle E/\mathbb{Q}$ to be a subfield of a cyclotomic field we require for $\displaystyle E/\mathbb{Q}$ to be a Galois extension with an abelian Galois group. This is obviously not true because the minimal polynomial of $\displaystyle \sqrt[3]{2}$ is $\displaystyle x^3 - 2$, and only one root of the three is contained in $\displaystyle E$.

By the way there is an incredible theorem in algebraic number theory. Above we said that for $\displaystyle E/\mathbb{Q}$ to be contained in a cyclotomic field we require for $\displaystyle E/\mathbb{Q}$ to be an (finite) abelian extension. However, the converse is true!!! In other words, if $\displaystyle E/\mathbb{Q}$ is a (finite) abelian extension then it is a subfield of some cyclotomic extension. This deep result is known as the Kronecker-Weber theorem.