Jacobi sums and cyclotomic matrices involving squares over finite fields

Let q=pⁿ be an odd prime power and let Fq be the finite field of q elements. Let Fq^{} be the group of all multiplicative characters of Fq and let be a generator of Fq^{}. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over Fq. For example, let s₁, s₂, , s (ₐ-₁) /₂ be the nonzero squares over Fq. For any integer 1 r q-2, define the matrix Bq (r): =ʳ (sᵢ+sⱼ) +ʳ (sᵢ-sⱼ) ₁ ₈, ₉ (ₐ-₁) /₂. We prove that if q 3 4, then (Bq (r) ) =₀ ₊ (ₐ-₃) /₂Jq (ʳ, ^2k) = cases (-1) ^q-3{4} iⁿGq (ʳ) ^q-1{2}/q & if\ r 1 2, Gq (ʳ) ^q-1{2}/q & if\ r 0 2. cases, where Jq (ʳ, ^2k) and Gq (ʳ) are the Jacobi sum and the Gauss sum over Fq respectively.