Marcus Number Fields Chapter 4 Exercise 8

Question

Let $r,e,f$ be given positive numbers. I should try to demonstrate that there are always $p,q$ prime such that $p$ splits exactly in $r$ different prime in the $q$th cyclotomic field.

I know that there are infinite $q$ such that $q-1$ is divisible by $r$, so I was trying to take such a prime and look for a correct value of $p$. I came into the problem of demonstrating that there is always a prime $p$ with order $(q-1)/r$ in $\mathbb{Z}/(q)^*$, wh9ich does not seem trivial to me.

Answer 1

Removing this from the unanswered queue.

• Let $q$ be a prime congruent to $1$ modulo $r$. Those exist by virtue of Dirichlet's theorem of primes in an arithmetic progression.
• Let $g$ be a primitive root modulo $q$. Let $a$ be the remainder of $g^r$ modulo $q$.
• Let $p$ be a prime congruent to $a$ modulo $q$. Such a prime exists again by Dirichlet.
• Because $p$ is of order $(q-1)/r$ we know that modulo $p$ the cyclotomic polynomial $\Phi_q(x)$ splits into exactly $r$ distinct irreducible factors, all of degree $(q-1)/r$. By basic algebraic number theory this means that the prime $p$ splits into a product of $r$ distinct prime ideals of $\Bbb{Z}[\zeta_q]$.