Prove each of the following. (A) Prove that if a and b are quadratic residues modulo m, then ab is also a quadratic residue mod m. (B) Prove that the square of a quadratic nonresidue of an odd prime p is a quadratic residue. (C) Prove that the product of a quadratic residue and a quadratic nonresidue of an odd prime p is a quadratic nonresidue.

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A: Since you have posted a multiple question according to guildlines I will solve first(a)   question…

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A: Given: 2+i To find:  The irr(ꭤ,ℚ) and deg(ꭤ,ℚ)for the given algebraic number ꭤ ∈ ℂ.

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A: Given:The integral To find:The integral using residue method.

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Q: Evaluate ·∞ In z de 4+x² 500 by the residue theorem: [*²].

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Q: Prove that if p and q are distinct odd primes, then we have (p-1)(q-1) 4 25 52 TID = (-1) quadratic…

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