Question

Please prove the 3 theorems, thank you!

7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p in any complete nesidue system modulo p are quadratic residuess modulo p and half are quadratic non-residues modulo p. From clementary school days, we have known that the product of a pos- itive number and a positive number is positive, a positive times a negative is negative, and the product of two negative numbers is positive. Quadratic residues and non-residues are related similarly. 7.7 Theorem. Suppose p is an odd prime and p does not divide either of the two integers a or b. Then 1. If a and b are both quadratic residues modulo p, then ab is a quadratic residue modulo p 2. Ifa is a quadratic residue modulo p and b is a quadratic non-residue modulo p, then ab is a quadratic non-residue modulo p: 3. If a and b are both quadratic non-residues mdulo p, then ab is a quadratic residue modulo p One of the mathematicians who studied quadratic residues modulo p was the Irench mathematician Legendre. He invented a symhol called the Legendre symbol that gives a value ofI to quadratic residues and - lo quadratic non-residues. The symbol is convenient because it lets us express theorems like the previous one in a compact way. Ilere is the definition. Definition. For an odd prime p and a natural number a with p not dividing (#) a, the Legendre symbol a is defined by if a is a quadratic residuc modulo p. ifa is a quadratic non-residuc modulo p. -1 Now we can express the preceding theorem using the Legendre symbol. 7.8 Theorem. Suppose p is an odd prime and p does not divide either a or b. Then ()(C) ab Our goal is to be ahlc to take an integer a and determine whether it is a quadratic residue modulo a prime p or a quadratic non-residue. Euler gave

Answer 1

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