Legendre symbol proof for (-5/p)

[abertram28]

I am working on some homework that I already handed in, but I can't get one of the problems. The fourth problem on the HW was to prove the forms of (-1/p), (2/p), (3/p), (-5/p), and (7/p).

I did this for -1 and 2 using the quadratic residues and generalizing a form for them. for 3 and 7 i used QRL, since they are both -1 mod 4, can i use QRL for the proof of -5 too? i know i got at least 80% on this problem, and that's a B+, so i should be fine on this problem. could someone please guide me on the first steps of this proof so that i can understand it? 3 and 7 were pretty easy, but I am not sure i got 7 right. most of it was in the book by David Burton that we use. BTW, I am a sophomore in math, so this class is really hard for me. that's why I am coming here for more understanding, that and my profs office hours are short and i use them for linear algebra.

for 3, i showed p congruent to 1 mod 4 for 4|p-1 and congruent to 1 mod 3 for 3|p-1, so 12|p-1, the forms of this p congruent to 3 mod 4 are 3 mod 12, 7 mod 12, 11 mod 12, and p congruent to 2 mod 3, if p congruent to 2 mod 12, 5 mod 12, 8 mod 12, 11 mod 12. the common solutions are p congruent to 1 and 11 mod 12, so its +- 1 mod 12, (3/p)=1, and since 8 is 0 mod 4, toss it, 3 and 9 are 0 mod 3, toss em, so 5,7 yield +- 5 mod 12, (3/p)=-1

can someone lead me through this for -5 now?

sorry for type settting, it wasnt really that necessary for this problem, and I am in a lab where i don't have much time left. sorry for long paragraphs too!

 

[shmoe]

First break it into cases depending on what (-1/p) is, which you should know everything about. Then use quadratic reciprocity to determine (5/p), according to what p is mod 5.

 

[abertram28]

ok, i see. I am needing to break (-5/p) into (5/p)(p/5)(-1/p) and solve for all the common congruences?

i get (-5/p) = {1 for p congruent to 1,9 (mod 20) and -1 for p congruent to -1,-9 (mod 20)}

is that right?

*EDIT* oops, don't i need to hit 3,7,13,17? *works on second half* *EDIT*

*2nd EDIT*

so, for p congruent to 3 (mod 20), both (-1/p) and (5/p) are -1, so 3 goes in the 1s, p congruent to 7 (mod 20), both (-1/p) and (5/p) are -1, so 7 goes in the 1s too, p congruent to 13, (-1/p) is 1, so its a -1s, same with 17...

so its (-5/p)={1 if p congruent to 1,3,7,9 (mod 20), -1 if p congruent to -1,-3,-7,-9 (mod 20)}

*2nd EDIT*

 

[prez]

what would the proof be if it was to be (-3/p)? i know it is suppose to end up as =1 if p == 1 mod 6, and -1 if p == -1 mod 6, but why?