Results 1 to 2 of 2

Thread: Quadratic reciprocity

  1. #1
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,115
    Thanks
    7

    Quadratic reciprocity

    Prove that $\displaystyle \displaystyle\sum\limits_{j=1}^{p-1} \left(\frac{j}{p}\right)=0$, p an odd prime.

    My attempt:

    If $\displaystyle p \equiv 3\ (mod\ 4)$,

    $\displaystyle \left(\frac{-j}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{j}{p}\right)=(-1)^{(p-1)/2}\left(\frac{j}{p}\right)=-\left(\frac{j}{p}\right)$ and the result follows immediately.

    But what if $\displaystyle p \equiv 1\ (mod\ 4)$?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jun 2010
    From
    Israel
    Posts
    148

    Re: Quadratic reciprocity

    Quote Originally Posted by alexmahone View Post
    Prove that $\displaystyle \displaystyle\sum\limits_{j=1}^{p-1} \left(\frac{j}{p}\right)=0$, p an odd prime.

    My attempt:

    If $\displaystyle p \equiv 3\ (mod\ 4)$,

    $\displaystyle \left(\frac{-j}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{j}{p}\right)=(-1)^{(p-1)/2}\left(\frac{j}{p}\right)=-\left(\frac{j}{p}\right)$ and the result follows immediately.

    But what if $\displaystyle p \equiv 1\ (mod\ 4)$?
    Theorem: For an odd prime $\displaystyle p$ there are exactly $\displaystyle (p-1)/2$ quadratic residues and exactly $\displaystyle (p-1)/2$ quadratic nonresidues.*

    So restating the theorem, half of the terms $\displaystyle (1/p), (2/p), ..., ((p-1)/p)$ are $\displaystyle -1$ and half are $\displaystyle 1$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Another quadratic reciprocity problem
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Jul 21st 2011, 05:04 PM
  2. Quadratic Reciprocity
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: May 20th 2009, 02:14 PM
  3. Quadratic Reciprocity Proof
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: Apr 17th 2009, 08:50 AM
  4. need help on quadratic reciprocity
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Apr 15th 2009, 01:01 PM
  5. Quadratic Reciprocity
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Sep 25th 2008, 09:09 AM

/mathhelpforum @mathhelpforum