Let me summarise what Hilbert says in his Zahlbericht about the behaviour of rational primes in the cyclotomic field $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $l$-th root of $1$ and $l$ is an odd prime. You can read the orginial at the Göttingen site or a French translation at the Grenoble site.
Satz 117. The ideal $\mathfrak{l}=(1-\zeta)\mathbb{Z}[\zeta]$ is prime (of residual degree 1), and $l\mathbb{Z}[\zeta]=\mathfrak{l}^{l-1}$.
Satz 118. The discriminant of the field $\mathbb{Q}(\zeta)$ is $(-1)^{(l-1)/2}l^{l-2}$.
Satz 119. If $p\neq l$ is a rational prime, $f>0$ the smallest exponent such that $p^f\equiv1\pmod l$, and $e$ is defined by $ef=l-1$, then we have $$p\mathbb{Z}[\zeta]=\mathfrak{p}_1\ldots\mathfrak{p}_e$$ where the $\mathfrak{p}_i$ are distinct prime ideals of residual degree $f$.
These results go back to Kummer (1847). All this was much before anyone dreamt of Class Field Theory.