Hi

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).

SK

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- Nov 6th 2010, 11:47 AMskykingPrime number problem
Hi

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).

SK - Nov 6th 2010, 12:29 PMroninpro
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. - Nov 6th 2010, 01:25 PMskyking
For this class I need to show this without complex numbers. We could use Fermat's litlle theoreme though but I can't see how

- Nov 6th 2010, 01:32 PMroninpro
This can also be done using the Euler criterion. Do you have it?

- Nov 6th 2010, 01:54 PMskyking
can't use it yet

- Nov 6th 2010, 03:04 PMAlso sprach Zarathustra
Let be a solution of . Therefor , from Fermat's theorem:

There is impossible that , because if then:

, and therefore , and the conclusion has , which is not true. Hence is from the form of .

The end!

By the way...

It also true that:

Let is a prime number, the equation has a solution__if and only if__ - Nov 8th 2010, 09:12 PMnerissa22
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