Verification: have I worked this legendre symbol correctly

Question

Compute $(\frac{92}{11}$).

Now $92^2 \equiv x^2\ mod(11) $ Now since there is no such x that satisfies this, so the legendre symbol is $-1$. Is this right?

Also, can somebody explain in a slightly more layman's language what the significance of quadratic residues are?

Answer 1

You can use the Legendre formula $\left( a \over p \right)=a^{(p-1)/2} \pmod p $ to manually compute the value. $\bmod 11$ you have: $$\left( 92 \over 11 \right)\equiv 92^5 \equiv 4^5 \equiv 4\times 16^2 \equiv 4\times 25 \equiv 100 \equiv 1$$ Therefor you value $-1$ is wrong.