Number of ways to represent an integer $n$ in a finite field $\mathbb F_{p^m}$ as a sum of two squares

Question

We know that in any finite field $\mathbb F$, any element $n \in \mathbb F$ can be written as a sum of two (integer) squares. I am wondering if there is an explicit formula of the number of ways to represent $n$ as a sum of two squares in $\mathbb F = \mathbb F_{p^m}$. I would be grateful for any references or insight.

Answer 1

A good reference is the book by Ireland and Rosen, "A Classical Introduction to Modern Number Theory. In chapter $8$ on Gauss and Jacobi sums they prove a theorem about the number of solutions of $$ a_1x_1^{l_1}+\cdots +a_rx_r^{l_r}=b $$ over a finite field. There is a case distinction for $b=0$ and $b\neq 0$, and the answer is in terms of $q$, Dirichlet characters $\chi_i$ and Jacobi sums $J(\chi_1,\ldots ,\chi_r)$. For example, if we denote by $N(f(x)=b)$ the number of solutions of $f(x)=b$ over $\Bbb F_p$, then $$ N(x_1^2+\cdots +x_r^2=1)=p^{r-1}+J(\chi,\ldots ,\chi), $$ where $\chi$ is a character of order $2$, the Legendre character.

Answer 2

Numerical calculations indicate that the number is independent of $n$ as long as $n\ne0$. In particular, given $n\in\Bbb F_q\setminus\{0\}$, the number of ordered pairs $(x,y) \in \Bbb F_q$ such that $x^2+y^2=n$ seems to be $q-\chi(q)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$.