Let $a \in \mathbb{Z}$ and consider the polynomial $f(X)=X^4+aX+1$; we assume that $a$ is chosen such that $f$ is irreducible and that the discriminant $4^4-27a^4=-p$ for some prime $p$ (for example $a=5$ gives $p=16619$). Let $L$ be the splitting field of $f$. We consider the map \begin{align} \mathcal{O}_L^{\times}/\left(\mathcal{O}_L^{\times}\right)^p \to \prod_{\mathfrak{p} | p} \mathcal{O}_{L, \mathfrak{p}}^{\times}/\left(\mathcal{O}_{L, \mathfrak{p}}^{\times}\right)^p, \end{align} where the product runs over primes $\mathfrak{p}$ of $\mathcal{O}_L$ dividing $p$.
Question: Is this map injective? When $a=5,21,25$ this is true, as can be checked by explicit computation.
If $\alpha$ is a root of $f$ in $L$, then it is enough to check this for $K=\mathbb{Q}(\alpha)$ by the Galois equivariance of the map. I am not aware of general techniques for studying this question (other than trying to write down generators of the unit group explicitly, which we haven't been able to do).
Context: The Galois group $\operatorname{Gal}(L/\mathbb{Q})$ is isomorphic to $S_4$ and is unramified outside $p,\infty$. Thus for $p \ge 5$ we get a Galois representation $G_{\mathbb{Q},\{p, \infty\}} \to \operatorname{GL}_3(\mathbb{F}_p)$, and I have a student trying to show it is neat in the sense of Mazur [1]. This would generalize the $S_3$ examples given in [1].
[1] B. Mazur, Deforming Galois representations, Galois groups over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172
