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.
Thanks,
Katy
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
Katy