MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

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