Let $\mathbb{F}_{p^{mn}}$ be a Finite Field with Char $p$ with $\zeta$ a primitive element in the field.
$\forall {k} \in $$\{$$1,\cdots,m$$\}$, $\forall {i}_{k} \in $$\{$$0,\cdots,p-1$$\}$, consider the two sets of $p^{m}$ elements of the form
$(1)$$\displaystyle \sum_{j=1}^{n} \displaystyle \sum_{k=1}^{m}(ji_{k} \mod p)p^{k+m(j-1)-1} \in \mathbb{Z}_{p^{mn}}$.
$(2)$$\displaystyle \sum_{j=1}^{n} \displaystyle \sum_{k=1}^{m}(ji_{k} \mod p)\zeta^{k+m(j-1)-1} \in \mathbb{F}_{p^{mn}}$.
$(3)$$\zeta^{\displaystyle \sum_{j=1}^{n} \displaystyle \sum_{k=1}^{m}(ji_{k} \mod p)p^{k+m(j-1)-1}} \in \mathbb{F}_{p^{mn}}$ where the exponent is in $\mathbb{Z}$.
Is there a name for these sets?
When $p$ is not prime we will not be in a finite field, however is there a term for $(1)$ in this case and is there something analogous to $(2)$ and $(3)$ in this case?
