I am stuck proving the theorem that there exists $x$, $x^4 \equiv 2 \pmod p$ iff $p$ is of the form $A^2 + 64B^2$.
So far I have got this (and I am not sure if it's correct)
Let $p = a^2 + b^2$ be an odd prime,
- $\left(\frac{a}{p}\right) = \left(\frac{p}{a}\right) = \left(\frac{a^2 + b^2}{a}\right) = \left(\frac{b^2}{a}\right) = 1$
since $p \equiv 1 \pmod 4$
- $\left(\frac{a+b}{p}\right) = \left(\frac{(a+b)^2-2ab}{a+b}\right) = \left(\frac{2}{a+b}\right) = (-1)^{((a+b)^2-1)/8}$
using the Jacobi symbol and second supliment of quadratic reciprocity.
- $(a+b)^{(p-1)/2} = (2ab)^{(p-1)/4}$
since $(a+b)^2 \equiv 2ab \pmod p$
and the last step which I'm stuck on now is for $p = a^2 + b^2$ let $x^2 \equiv -1 \pmod p$ then $2^{(p-1)/4} = x^{ab/2}$. And I don't see how to prove the theorem with this result.
We can asume that 2 is a quadratic residue mod $p$ and so that $p \equiv 1 \pmod 8$ and this implies that if we pick $a$ odd and $b$ even then $b$ is a multiple of 4. We have to prove that $b$ is a multiple of 8.
First observe that as $x^2 \equiv -1 \pmod{p}$ and $a^2 + b^2 = p$ we have $$ \left(\frac{a+b}{p}\right) \equiv (-1)^{((a+b)^2-1)/8} \equiv x^{(p+2ab-1)/4} \pmod{p} $$ note that the exponent of $x$ is even because $b$ is multiple of 4, so we can chose the sign of the base as we wish.
In adition we also have $$ -a^2 \equiv b^2 \pmod{p} $$ so $(xa)^2 \equiv b^2 \pmod{p}$ and $$ax \equiv \pm b \pmod{p}$$ and picking the sign of $x$ we can asume that $ax \equiv b \pmod{p}$. So $ab \equiv a^2 x$ and $$ (ab)^{(p-1)/4} = a^{(p-1)/2} x^{(p-1)/4} \equiv x^{(p-1)/4} \pmod{p} $$ because $a$ is a quadratic residue mod $p$.
The identity you obtained: $$\left(\frac{a+b}{p}\right) = (a+b)^{(p-1)/2} \equiv (2ab)^{(p-1)/4} \pmod p $$ now becomes $$x^{(p+2ab-1)/4} \equiv 2^{(p-1)/4} x^{(p-1)/4} \pmod p $$ and in consequence $$2^{(p-1)/4} \equiv x^{ab/2} \pmod p $$ As $b$ is multiple of 4 say $b = 4b'$, we have $$2^{(p-1)/4} \equiv (-1)^{ab'} \pmod p $$ so $2$ is a biquadratic residue iif $b'$ is even or what is the same thing iif $b$ is multiple of 8 and we are done.