Hi, I have this problem.
Let p be an odd prime. Show that the legendre symbol (5/p)= 1 if and only if .
I am really struggling to understand this legendre symbol stuff so if anyone could give me a clue I would be very grateful.
You've already done the hard work. Once you know that
you just have to look at the quadratic residues mod 5. Now any number, not just a prime p, is equivalent to 1,2,3 or 4 mod 5 (we ignore numbers divisible by 5). So we test them 1 by 1.
So we've tested all possible numbers and found that 1 and -1 are the only quadratic residues, and that 2 and 3 are not quadratic residues for any number. It's as simple as that.
Yes your right I meant to write mod 5. Sorry.
So what do you do when you cannot know if or ?
For example . You dont know if is +1 or -1.
Do you just look at both cases? Because I am suppose to prove that
I dont understand where the (mod 28) comes from here. I really dont understand this stuff very well.
Thank you for all your help