Complete splitting criterion for a cyclotomic field

Given: The cyclotomic field $K=\mathbb{Q}({\zeta }_f)$ presented by its conductor $f$ (Cyclotomic conductor of a full cyclotomic field), and a rational prime $\mathrm{\ell }\nmid f$, so $\gcd (\mathrm{\ell },f)=1$ and the class of $\mathrm{\ell }$ lies in $(\mathbb{Z}\mathrm{/}f{)}^{\times }$.

Unramified decomposition: for the conductor $f$ and $\mathrm{\ell }\nmid f$, every prime of ${\mathcal{O}}_K$ above $\mathrm{\ell }$ has residue degree ${\mathrm{ord}}_f(\mathrm{\ell })$ and there are exactly $\phi (f)\mathrm{/}{\mathrm{ord}}_f(\mathrm{\ell })$ of them; in particular $\mathrm{\ell }$ is unramified (Decomposition of an unramified prime in a cyclotomic field).

Splitting terminology: a rational prime $\mathrm{\ell }$ splits completely in $K$ if it is unramified in $K$ and every prime of ${\mathcal{O}}_K$ above it has residue degree $1$ (Splitting and ramification terminology).

For a positive integer $f$ and an integer $\mathrm{\ell }$ with $\gcd (\mathrm{\ell },f)=1$, the order of the class $[\mathrm{\ell }]\in (\mathbb{Z}\mathrm{/}f{)}^{\times }$ equals $1$ if and only if $\mathrm{\ell }\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}f)$ (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity, The unit group $(\mathbb{Z}\mathrm{/}n{)}^{\times }$ and Euler's totient $\phi (n)=\mid (\mathbb{Z}\mathrm{/}n{)}^{\times }\mid$ for $n\ge 1$).

$[K:\mathbb{Q}]=\phi (f)$ ($[\mathbb{Q}({\zeta }_n):\mathbb{Q}]=\phi (n)$ and $\mathrm{Gal}(\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q})\cong (\mathbb{Z}\mathrm{/}n{)}^{\times }$).