Can anyone point me in the direction of a proof for the generalisation of the Quadratic Reciprocity Law to the Jacobi Symbol?

Alternatively give me some pointers on how to prove it myself.

Printable View

- Sep 24th 2008, 03:26 AMKiwi_DaveQuadratic Reciprocity
Can anyone point me in the direction of a proof for the generalisation of the Quadratic Reciprocity Law to the Jacobi Symbol?

Alternatively give me some pointers on how to prove it myself. - Sep 24th 2008, 06:26 AMThePerfectHacker
- Sep 24th 2008, 09:20 PMKiwi_Dave

Thank's for the help. I can see the truth of the final congruence and will write my argument down. But is there a more susinct argument?

Each summand on the LHS contributes to the sum (mod 4) if none of the following statements are true:

(mod 4)

(mod 4)

That is they contribute iff

(mod 4)

And the sum is equal to the number of contributions (mod 2). The number of p_j such that (mod 4) and the number of q_i such that (mod 4) must both be odd or the sum is zero (mod 2).

Now

where r is the number of p_j that equal 3 (mod 4). Therefore, if there are an even number of p_j that equal 3 (mod 4) and otherwise. In conclusion the sum is only non zero if .

Similarly the sum is only non zero if .

Now if or . Conversely (mod 2) if and . - Sep 25th 2008, 10:09 AMThePerfectHacker
If are odd then . Proof: thus and so . Thus, .

Now generalize this by induction for i.e. .

Thus, .