Quadratic Gauss sums on matrices (Q1827490)

Let \(F = F_{ p^\alpha }\) be a finite field of order \(p^\alpha\) for some prime \(p\). The author defines a Gauss sum on matrices \(A \in M_n(F)\) as follows: \[ G_s(A) := \sum_{ X \in M_n(F) } \exp \left( {{ 2 \pi i } \over { p }} \, \text{tr}_F \left( \text{tr} \left( A X^s \right) \right) \right) , \] where \(\text{ tr}_F\) denotes the trace map of field extensions \(F/F_p\) and the second tr on the right-hand side is the trace map on matrices. The author derives a formula for \(G_2(A)\) for any \(A \in M_n(F)\) in terms of the usual Gauss sum \[ g_s (a) := \sum_{ x \in F } \exp \left( {{ 2 \pi i } \over { p }} \, \text{tr}_F \left( ax^s \right) \right) . \]