I know that 2 is a residue of primes of the form $8n+1$ and $8n+7$ and so on. I want to find a purely group theoretic or field theoretic proof of these statements.
For example, for 8n+1, the multiplicative group is of the order of 8 and so there exists an element of order 8, j. Then $(j + 1/j)^2 = 2$. Similarly for $8n+5$, there is an element of order 4, $i$ and if 2 had a residue, we could construct j such that $j^2 = i$ and therefore there is an element of order 8 which is impossible.
What I am not getting stuck on is finding something that differentiates a field of the order $8n+7$ from something of the form $8n+3$. This is how I proved the last 2 cases(there either exists or does not exist something of order 8).