**Q** is the rational number field.
p is a prime number.
q is a prime number other than p.
$k\_{p^r}$ is a cyclotomic field.
$k\_{p^r}$=**Q**(x) where x is exp(2$\pi$i/$p^r$).
[$k\_{p^r}$:**Q**]=$p^{r-1}(p-1)$.


Question: Does q remain a prime in the integer ring of $k\_{p^r}$?



Cyclotomic Fields over Q and prime ideals 
Theorem I.2.13 of Washington's book on cyclotomic fields says the following: $K$ is the $n$th cyclotomic field and $p\nmid n$, let $f$ be the smallest positive integer such that $p^f\equiv 1 (\mathrm{mod}~n)$. Then $p$ splits into $\phi(n)/f$ distinct primes in $K$.


