Of interest to me is the following question (it would be nice to find out what is known in its direction):

given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime number $$p = \prod_{\sigma \in Gal(K/\mathbb{Q})}{\sigma(q)},$$

where $q \in \mathcal{O}_K$ is a prime element, "how probable", in a loose sense, is that the images of the numbers $\sigma(q)$, $(\sigma \neq id)$ in the residue field $\mathcal{O}_K/q\mathcal{O}_K$ $(\cong \mathbb{F}_p)$ have orders of one's choice? For instance, can one always find "many" $p$ such that, for any choice of distinct elements ${\sigma_1, \sigma_2, ... \sigma_r \in Gal(K/\mathbb{Q})}$, the numbers $\sigma_1(q), ..., \sigma_r(q) \in Gal(K/\mathbb{Q})$ were quadratic residues in the field $\mathcal{O}_K/q\mathcal{O}_K$, and the remaining numbers $\sigma(q), \sigma \neq \sigma_j, j = 1, ..., r, \sigma \neq id$ - quadratic nonresidues?

Of interest to me is the following question (it would be nice to find out what is known in its direction):

Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime number $$p = \prod_{\sigma \in \mathrm{Gal}(K/\mathbb{Q})}{\sigma(q)},$$

where $q \in \mathcal{O}_K$ is a prime element, "how probable", in a loose sense, is it that the images of the numbers $\sigma(q)$, $(\sigma \neq \mathrm{id})$ in the residue field $\mathcal{O}_K/(q)$ $(\cong \mathbb{F}_p)$ have multiplicative orders of one's choice? For instance, given some choice of distinct elements ${\sigma_1, \sigma_2, \ldots, \sigma_r \in \mathrm{Gal}(K/\mathbb{Q})}$, can one always find "many" $p$ such that the numbers $\sigma_1(q), \ldots, \sigma_r(q) \in \mathrm{Gal}(K/\mathbb{Q})$ are quadratic residues in the field $\mathcal{O}_K/(q)$, and the remaining numbers $\sigma(q),\; \sigma \neq \sigma_j\; (j = 1, \ldots, r),\; \sigma \neq \mathrm{id}$ are quadratic nonresidues?

Of interest to me is the following question (it would be nice to find out what is known in its direction):

given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime number $$p = \prod_{\sigma \in Gal(K/\mathbb{Q})}{\sigma(q)},$$

where $q \in \mathcal{O}_K$ is a prime element, "how probable", in a loose sense, is that the images of the numbers $\sigma(q)$, $(\sigma \neq id)$ in the residue field $\mathcal{O}_K/q$ $(\cong \mathbb{F}_p)$ have orders of one's choice. For instance, can one always find "many" $p$ such that, for any choice of distinct elements ${\sigma_1, \sigma_2, ... \sigma_r \in Gal(K/\mathbb{Q})}$, the numbers $\sigma_1(q), ..., \sigma_r(q) \in Gal(K/\mathbb{Q})$ were quadratic residues in the field $\mathcal{O}_K/q$, and the remaining numbers $\sigma(q), \sigma \neq \sigma_j, j = 1, ..., r, \sigma \neq id$ - quadratic nonresidues?

Of interest to me is the following question (it would be nice to find out what is known in its direction):

given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime number $$p = \prod_{\sigma \in Gal(K/\mathbb{Q})}{\sigma(q)},$$

where $q \in \mathcal{O}_K$ is a prime element, "how probable", in a loose sense, is that the images of the numbers $\sigma(q)$, $(\sigma \neq id)$ in the residue field $\mathcal{O}_K/q\mathcal{O}_K$ $(\cong \mathbb{F}_p)$ have orders of one's choice? For instance, can one always find "many" $p$ such that, for any choice of distinct elements ${\sigma_1, \sigma_2, ... \sigma_r \in Gal(K/\mathbb{Q})}$, the numbers $\sigma_1(q), ..., \sigma_r(q) \in Gal(K/\mathbb{Q})$ were quadratic residues in the field $\mathcal{O}_K/q\mathcal{O}_K$, and the remaining numbers $\sigma(q), \sigma \neq \sigma_j, j = 1, ..., r, \sigma \neq id$ - quadratic nonresidues?