# the legendre symbol

• Nov 3rd 2008, 12:22 PM
mndi1105
the 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 PM
chiph588@
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?