1. (15 points) For each of the following statements, indicate whether the statement is true or false and justify your an

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1 15 Points For Each Of The Following Statements Indicate Whether The Statement Is True Or False And Justify Your An 1 (123.14 KiB) Viewed 56 times

1. (15 points) For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) Let p be an odd prime. If α,β∈Fp​ are nonsquares, then αβ is a square. (b) If m∈N with m≥3, then the product of the elements in (Z/mZ)×is congruent to −1 modulo m. (c) The equation X2−13Y2=7 has an integral solution, i.e., there is a pair (x,y)∈Z2 such that x2−13y2=7 (d) If p is prime and β∈Fp×​, then the equation Xp−X=β has no solutions in Fp​.