= Solution
A <cyclotomic field> over $\mathbb Q$ is a field $\mathbb Q(\zeta_m)$ generated by an $m$th <root of unity> of exact order $m$. It contains all roots of $X^m-1$, and is its <splitting field>. In characteristic zero this <polynomial> is separable, so the extension is Galois.
Every automorphism sends $\zeta_m$ to $\zeta_m^a$ with $a\in(\mathbb Z/m\mathbb Z)^\times$, and is determined by that image. Composition multiplies the exponents modulo $m$, yielding an injective homomorphism into an <abelian group>. Thus \b[the <Galois group> is abelian]. Using irreducibility of the <cyclotomic polynomial>, every unit exponent occurs and
$$
\boxed{\operatorname{Gal}(\mathbb Q(\zeta_m)/\mathbb Q)\cong(\mathbb Z/m\mathbb Z)^\times.}
$$
If “cyclotomic” is used more broadly for a subfield of such a field, it is still Galois and abelian, because every subgroup of an abelian <Galois group> is normal.
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