Take the 2-minute tour ×  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).
share|improve this question Is that a homework? –  Mark Sapir
May 31 '13 at 7:38 Voted to close. –  Mark Sapir May 31 '13 at
8:59 The title asks about $-2$, the body doesn't. Anyway, voting
to migrate to m.se –  Gerry Myerson Jul 19 '13 at 0:26 add
comment  migrated fr