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 original 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, $f>0$ is 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.

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 original 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, $f>0$ is the smallest exponent such that $p^f\equiv1\pmod l$, and $e$ is defined by $ef=l-1$, then $$ p\mathbb{Z}[\zeta]=\mathfrak{p}_1\ldots\mathfrak{p}_e, $$ 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.

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.

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 original 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, $f>0$ is 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.