Results 1 to 2 of 2

Math Help - Law of Quadratic reciprocity2

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    46

    Law of Quadratic reciprocity2

    Let p be an odd prime number. Prove that

    (3 over p) =1 if and only if p = +-1 mod 12

    where (3 over p) denotes the legendre symbol.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by mndi1105 View Post
    Let p be an odd prime number. Prove that

    (3 over p) =1 if and only if p = +-1 mod 12

    where (3 over p) denotes the legendre symbol.
    Let p>3. There are two cases: p\equiv 1(\bmod 4) or p\equiv 3(\bmod 4). In the first case we get (3/p) = (p/3). Now if p\equiv 1(\bmod 3) then (p/3) = (1/3)=1 and if p\equiv 2(\bmod 3) then (p/3) = (2/3) = -1, therefore p\equiv 1(\bmod 3) in the first case. Together p\equiv 1(\bmod 4) and p\equiv 1(\bmod 3) give us p\equiv 1(\bmod 12).

    In the second case we get (3/p) = -(p/3) and to get (3/p) = 1 it is necessary and sufficient to get (p/3) = -1. Now this happens when p\equiv 2(\bmod 3). We have p\equiv 3(\bmod 4) and p\equiv 2(\bmod 3) which is equivalent to p\equiv -1(\bmod 4) and p\equiv -1(\bmod 3). Together this combines into p\equiv -1(\bmod 12).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quadratic help
    Posted in the Algebra Forum
    Replies: 2
    Last Post: April 11th 2010, 05:42 PM
  2. quadratic?
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 10th 2010, 05:05 AM
  3. Replies: 10
    Last Post: May 6th 2009, 09:52 AM
  4. Replies: 1
    Last Post: June 12th 2008, 09:30 PM
  5. Quadratic Eq.
    Posted in the Algebra Forum
    Replies: 2
    Last Post: May 27th 2008, 06:54 PM

Search Tags


/mathhelpforum @mathhelpforum