Given $d \in \mathbb{Z}^+$ and $a \in \mathbb{Q}^*$, let $K = \mathbb{Q}(\zeta_d, a^{1/d})$, where $\zeta_d$ is a primitive $d$th root of unity.
I'm trying to prove that: "A prime number $p$ splits completely in $K$ if and only if $p \equiv 1 \pmod d$ and $a$ is a $d$th power modulo $p$"
I tried to use Dedekind-Kummer factorization theorem (the one that relates the decomposition of a prime $p$ with the factorization of a polynomial modulo $p$) but the reasoning seems quite complicate. I guess that there's a more clever method.
Thank for any help
Notes: (1) It might be necessary to exclude finitely many primes from the statement of the criterion, like the one dividing the numerator or denominator of $a$; (2) Since $a$ is a rational number, by $d$th power modulo $p$ I mean that there exists $b \in \mathbb{Q}$ such that the numerator of $a - b^d$ is divisible by $p$.