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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 be the unique quadratic multiplicative character of the finite field Fq. If set \s₁, , s (ₐ-₁) /₂\=\x²: \ xq\{0\\}, then we prove that + (sᵢ+sⱼ) + (sᵢ-sⱼ) ₁ ₈, ₉ (ₐ-₁) /₂= (q-12t-1) q^q-3{4}. This confirms a conjecture of Zhi-Wei Sun.
Li et al. (Tue,) studied this question.
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