"Consecutive" square residues in odd-order finite fields

Question

Let $\mathbb F = \mathrm{GF}(p^r)$ be finite field for $p$ an odd prime, and define $$ \begin{align} Q &= \bigl\{ u^2 \,\big|\, u \in \mathbb F^\times \bigr\} \\ N &= \mathbb F^\times \smallsetminus Q \end{align} $$ As $\mathbb F^\times$ is a cyclic group of even order, we have $\lvert Q \rvert = \lvert N \rvert = \tfrac{1}{2}(p^r - 1)$. A classic result of number theory for the case $r = 1$ is that $$ \# \bigl\{ q \in Q \;\big|\; q+1 \in Q \bigr\} = \begin{cases} (p-5)/4, & \text{if $p \equiv 1 \pmod{4}$}; \\[1ex] (p-3)/4, & \text{if $p \equiv 3 \pmod{4}$}. \end{cases} $$ Is there a corresponding result for $r > 1$ as well, and how is it shown?

Answer 1

We can in fact adapt a standard proof of the result for $r = 1$, such as the one presented in this answer to the related question for the integers modulo $p$. Below, let $\chi: \def\F{\mathbb F}\F \to \{-1,0,+1\}$ be the quadratic character of $\F^\times$ extended to the whole field, so that $$ \chi(a) = \begin{cases} +1 & \text{for $a \in Q$}, \\ -1 & \text{for $a \in N$}, \\ 0 & \text{for $a = 0$}. \end{cases} $$ Let $C = \# \bigl\{ q \in Q \,\big|\, q+1 \in Q \bigr\}$: then we have $$ C = \sum_{a \in \F}\Bigl(\frac{1+\chi(a)}2 \Bigr)\Bigl(\frac{1+\chi(a{+}1)}{2}\Bigr) - \frac{1}{2} - \Bigl(\frac{1 +\chi(-1)}{4}\Bigr), $$ where the first subtracted term is to correct for the $a = 0$ contribution, and the second subtracted term is to correct for the $a = -1$ contribution. Since $\lvert N \rvert = \lvert Q \rvert$, we have $$ \sum_{a \in \F} \chi(a) = \sum_{a \in \F} \chi(a+1) = 0; $$ then because $\chi$ is a multiplicative homomorphism, and as $a^{-1} \in Q \iff a \in Q$, we have
$$\begin{align} C &= \frac{\lvert \F \rvert}{4} + \frac{1}{4}\sum_{a \in \F} \chi(a) \chi(a+1) -\frac{1}{2} - \Bigl(\frac{1 + \chi(-1)}{4}\Bigr) \\&= \frac{1}{4}\biggl[\lvert \F \rvert + \sum_{a \in \F^\times} \chi\bigl(a^{-1}(a+1)\bigr) - 2 - \Bigl(1 + \chi(-1)\Bigr)\biggr] \\&= \frac{1}{4}\biggl[ p^r - 3 - \chi(-1) + \sum_{a \in \F^\times} \chi\bigl(1+a^{-1}\bigr) \biggr] \\&= \frac{1}{4}\biggl[ p^r - 3 - \chi(-1) + \!\sum_{\! b \in \F \smallsetminus \{1\}\!\!} \chi(b) \biggr] \\&= \frac{1}{4}\Bigl[ p^r - 4 - \chi(-1) \Bigr], \end{align} $$ which is precisely the same as for the case as $r=1$ (in which $\chi(-1) = +1$ for $p \equiv 1 \pmod{4}$ and $\chi(-1) = -1$ otherwise) except with an included exponent. In particular: as $$ \chi(-1) = \begin{cases} +1 &\text{if $\F\:\!{:}\:\!\mathbb Z_p$ is an even degree extension or $p \equiv 1 \pmod{4}$}; \\ -1 &\text{otherwise}, \end{cases} $$ we have $\chi(-1) = +1$ if $p^r \equiv 1 \pmod{4}$, and $\chi(-1) = -1$ otherwise. Thus we have $$ C = \begin{cases} (p^r-5)/4, & \text{if $p^r \equiv 1 \pmod{4}$}; \\[1ex] (p^r-3)/4, & \text{if $p^r \equiv 3 \pmod{4}$}. \end{cases} $$