I am trying to understand how primes ramify in $\mathbb Q(\zeta _p)/\mathbb Q$ for $p$ prime. From class field theory, how a prime $q$ ramifies depends only on $q \mod p$.
I have following particular question in mind:
Which primes have inertia degree $f=6$ in $\mathbb Q (\zeta _{31}) /\mathbb Q?$
We know that $(31)$ totally ramifies and all other primes are unramified. For these unramified primes, their Frobenius element generates the Decomposition group which has size $f$ (as $e=1$). Frobenius of $q$ acts as $\zeta _{31} \mapsto \zeta _{31} ^q$. So we look for $q$ such that $\zeta _{31} ^{q^6} =1$ i.e. primes such that $q^6 =1 \mod 31$. Finding solutions to $x^6=1\mod 31$ seems difficult though with help of computer I have found that elements of order $6$ in $(\mathbb Z/31 \mathbb Z)^\times$ are $6$ and $26$. So primes which are $6 \text{ or } 26 \mod 31$ have inertia degree in $\mathbb Q (\zeta _{31}) /\mathbb Q$. Some examples are $37,181,223,347,367,409,491$.
But this seems very difficult as it requires dealing with $(\mathbb Z /p \mathbb Z)^\times$. Is there any way to simplify my work above?
I was told that this question can be answered using Dirichlet characters. How do Dirichlet characters help us? I know that we need to find primes such that $$\frac{|\{\chi : (\mathbb Z /p \mathbb Z)^\times \rightarrow \mathbb C ^\times \ | \ \chi (q) \neq 0\}|}{| \{\chi : (\mathbb Z /p \mathbb Z)^\times \rightarrow \mathbb C ^\times \ | \ \chi (q) = 1\}|}=6 $$
but this looks difficult too for it requires knowledge of the Dirichlet characters group.
Is there an easy answer to this question and can I generalize?
Please feel free to provide any reference, thank you in advance.