= Solution
The <cyclotomic polynomial> is
$$
\Phi_n(X)=\prod_{\substack{1\le a\le n\\\gcd(a,n)=1}}(X-\xi^a).
$$
By the allowed irreducibility, this is the <minimal polynomial> of $\xi$ and its degree is $\varphi(n)$. The <field> $\mathbb Q(\xi)$ contains every $n$th <root of unity> and is the <splitting field> of $X^n-1$. This <polynomial> has no repeated roots in characteristic zero, so the extension is Galois.
An <automorphism> sends $\xi$ to another primitive root $\xi^a$, where $a$ is a unit modulo $n$. Conversely each such conjugate defines an embedding, and it is an <automorphism> of $\mathbb Q(\xi)$ because $a$ has an inverse modulo $n$. Composition corresponds to multiplication of exponents. Therefore
$$
\boxed{\operatorname{Gal}(\mathbb Q(\xi)/\mathbb Q)\cong(\mathbb Z/n\mathbb Z)^\times.}
$$
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