A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields

Motivated by the recent work of Zhi-Wei Sun on determinants involving the Legendre symbol, in this paper, we study some matrices concerning subgroups of finite fields. For example, let q≡ 3 4 be an odd prime power and let φ be the unique quadratic multiplicative character of the finite field Fq. If set ₁,⋯,s(q-1)/2\=²:\ xq\0\\, then we prove that [t+φ(sᵢ+sⱼ)+φ(sᵢ-sⱼ)]1≤ i,j≤ (q-1)/2=(q-1/2t-1)qq-3/4. This confirms a conjecture of Zhi-Wei Sun.