let p be an odd prime number and let a, b be integers with p does not divide a nor b. Prove that among the congruences x^2= a mod p, x^2= b mod p, and x^2= ab mod p either all three are solvable or exactly one is solvable.

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- Nov 3rd 2008, 12:22 PMmndi1105the legendre symbol
let p be an odd prime number and let a, b be integers with p does not divide a nor b. Prove that among the congruences x^2= a mod p, x^2= b mod p, and x^2= ab mod p either all three are solvable or exactly one is solvable.

- Nov 3rd 2008, 02:34 PMchiph588@
Note

So now, if , then

if , then

now WLOG assume and , then

Can you see now?