= Solution
By <Dirichlet theorem on primes in arithmetic progressions>, choose a prime $\ell$ with
$$
\ell\equiv1\pmod r.
$$
The <cyclotomic field> $E=\mathbb Q(\zeta_\ell)$ is Galois over $\mathbb Q$, and
$$
\operatorname{Gal}(E/\mathbb Q)
\cong(\mathbb Z/\ell\mathbb Z)^\times
\cong C_{\ell-1},
$$
where the last group is cyclic because the multiplicative group of a finite field is cyclic.
The cyclic group $C_{\ell-1}$ has a unique subgroup $H$ of order $(\ell-1)/r$. By the <Galois correspondence>, its fixed field $L=E^H$ satisfies
$$
\operatorname{Gal}(L/\mathbb Q)
\cong \operatorname{Gal}(E/\mathbb Q)/H
\cong C_r.
$$
Normality follows because $H$ lies in an abelian group. Thus every $r>1$ occurs as the Galois group of a finite extension of $\mathbb Q$. This is the <cyclic Galois extension of the rational numbers of every finite degree>.
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