I need direction with the following problem.
p<>2 is a prime number, suppose the equation x^2=-1(modp) has a solution.
Show that p=1(mod4).
You could use complex numbers to prove this.
First note that if , then is prime. Now it will suffice to show that if has a solution, then is not prime. (This will force .
If the equation is satisfied, then there exists an integer such that . We can factor the right hand side to receive . If were prime, then or , but this would mean that , which is absurd. Therefore, is not prime.
There is impossible that , because if then:
, and therefore , and the conclusion has , which is not true. Hence is from the form of .
By the way...
It also true that:
Let is a prime number, the equation has a solution if and only if