Splitting in cyclotomic field $Q(\zeta_5)$

Question

My question is about the behavior of primes $p\equiv 2,3,4 \bmod 5$ in $Q(\zeta_5)$ if they are inert or splitting in 2 primes or 4 primes??

Answer 1

In general, if $p$ is a prime not dividing $n$, the number of primes in the cyclotomic field $\Bbb Q(\zeta_n)$ dividing $p$ is $\phi(n)/k$ where $\phi$ is the Euler function and $k$ is the multiplicative order of $p$ modulo $n$.

In the case $n=5$, if $p\equiv1\pmod5$, $k=1$ and $p$ splits into four ideal factors, if $p\equiv4\pmod 5$ then $k=2$ and $p$ splits into two ideal factors, and if $p\equiv2$ or $3\pmod5$ then $k=4$ and $p$ is inert.