norme of element in cyclotomic fields

Question

let $\pi$ a prime in $Z[\zeta_5]$ over a prime $p$ in $Z$. my quesyion is : can we write the norme $N_{Q(\zeta_5)/Q}$ in term of $p$ ?????

Answer 1

It's $p^f$ where $f$ is the inertial degree of the prime ideals of $\Bbb Z[\zeta_5]$ over $p$. This is the least positive integer with $p^f\equiv1\pmod 5$. See my previous reply for more about this $f$.