3 problems

Fix an integer $d\ge 2$. For each odd prime power q\equiv1{\ifmmode\text{\rm\ (mod~d)}\else\discretionary{}{}{\text{ }}\rm(mod~d)\fi}, let $r_q$ be the cardinality of a maxim…

Let $d>1$ be an integer. Let $q \equiv 1 \pmod {2d}$ be a power of a prime $p$, and let $r$ be the largest integer such that $d \mid \frac{q-1}{p^r-1}$. The subfield…

Let $d$ be a positive integer greater than $1$. Let $q \equiv 1 \pmod {2d}$ be a power of a prime $p$, and let $r$ be the largest integer such that $d \mid \frac{q-1}{p^r-1}$. The…