43. State the value of the Legendre symbol ) where p is an odd prime, and...

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Answer 1

Answer #1

Theorem : If p is an odd prime, then (-1/p) = 1 if p ≡ 1 (mod 4), and (-1/p) = -1 if p ≡ 3 (mod 4)

Proof : Using Euler's Criterion, we know that,

(−1/p) ≡ (−1)^(p-1)/2 (mod p).

Suppose that, p ≡ 1(mod 4). That is p = 3, 7, 11, …. When p is any of these values, then (p−1)/2 is even, and hence (−1/p) ≡ 1(mod 4).

Suppose that, p ≡ 3(mod 4), that is p = 5, 13, 19, …, then (p−1)/2 is odd, and hence (−1/p) ≡ −1(mod p). Hence our proof is complete.

Answer 2

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