the legendre symbol

**mndi1105** · November 3rd 2008, 12:22 PM

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.

**chiph588@** · November 3rd 2008, 02:34 PM

Note $\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$

So now, if $\left(\frac{a}{p}\right) = \left(\frac{b}{p}\right) = 1$ , then $\left(\frac{ab}{p}\right) = 1$

if $\left(\frac{a}{p}\right) = \left(\frac{b}{p}\right) = -1$ , then $\left(\frac{ab}{p}\right) = 1$

now WLOG assume $\left(\frac{a}{p}\right) = 1$ and $\left(\frac{b}{p}\right) = -1$ , then $\left(\frac{ab}{p}\right) = -1$

Can you see now?

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