let give a prime number p>2 and a /D(a,p)=1 and suppose that the equation x^2 ≡ a[p]have a solution1) 1) prove that a^((p−1)/2) ≡ 1 [p] 2)prove that x^2 ≡ −1[p] ⇔ p≡ 1 [4]


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Question Number 29035 by abdo imad last updated on 03/Feb/18

$l e t g i v e a p r i m e n u m b e r p > 2 a n d a / D (a, p) = 1 a n d$ $s u p p o s e t h a t t h e e q u a t i o n x^{2} \equiv a [p] h a v e a s o l u t i o n 1)$ $1) p r o v e t h a t a^{\frac{p - 1}{2}} \equiv 1 [p]$ $2) p r o v e t h a t x^{2} \equiv - 1 [p] \Leftrightarrow p \equiv 1 [4]$
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