Show that if $a$ is a quadratic residue $(\mod{p})$ and $ab \equiv 1 (\mod{p})$ then $b$ is a quadratic residue $(\mod{p})$.

I am not sure where to go with this proof, I started with:
$x^2 \equiv a (\mod{p})$ because $a$ is a quadratic residue.
$x^2b \equiv ab (\mod{p})$
$x^2b \equiv 1 (\mod{p})$

Stumped right here.

Then I thought maybe using Legendre's symbols would be better.
$(a/p) = 1$ because $a$ is a quadratic residue

So I have two starting points and no finish line in sight.

Thanks in advance for any help.

But not sure where to go to show that $(b/p) = 1$

2. ## Re: Quadratic Residue proof

You do know the multiplicative property of the Legendre symbol, right? (That $(a/p)(b/p)=(ab/p)$).

Also, $(1/p)=1$.

3. ## Re: Quadratic Residue proof

Ah I see now. Thanks!

$(a/p) = 1$
$(a/p)*(b/p) = 1*(b/p)$
$(ab/p) = (b/p)$
$(1/p) = (b/p)$
$1 = (b/p)$