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Math Help - Extended quadratic symbol

  1. #1
    Newbie
    Joined
    Dec 2009
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    4

    Extended quadratic symbol

    Hello. I have a few questions about Legendre and extended quadratic legendre symbols.

    I understand the definition of the legendre symbol, but when it comes to the extended or Jacobi symbol I get a little confused. It says that:

    For arbitrary integers n which factor as

    n=(2^e(0))*(P(1)^e(1))*....*(P(k)^e(k)) the jacobi symbol is defined as
    (b/n)=[(b/P(1))^e(1)]*[(b/P(2))^e(2)]....*[(b/P(k))^e(k)].



    In our notation, those e's are sometimes just positive numbers, and other times they are the identity element, but it doesn't define it here. If it is the identity element, could someone elaborate where they come into play?


    For an example, is it correct to say:

    if n=pq where p and q are distinct primes, and (a/n)=-1,

    then (a/n)=(a/p)*(a/q)=-1?

    Sorry for the cramped notation, I've seen people use the proper notation on these forums, but I don't know how to do it

    Thanks in advance.
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  2. #2
    Senior Member
    Joined
    Dec 2007
    From
    Melbourne
    Posts
    428

    In our notation, those e's are sometimes just positive numbers, and other times they are the identity element, but it doesn't define it here. If it is the identity element, could someone elaborate where they come into play?
    In your first expression
    n=(2^e(0))*(P(1)^e(1))*....*(P(k)^e(k))
    all of the P s are prime, so the e s are the powers of each prime in the prime factorisation of n.


    For an example, is it correct to say:

    if n=pq where p and q are distinct primes, and (a/n)=-1,

    then (a/n)=(a/p)*(a/q)=-1?
    yes

    Sorry for the cramped notation, I've seen people use the proper notation on these forums, but I don't know how to do it
    If you know latex, you can access the displaymath enviroment by entering or clicking the button with \Sigma on it. If you don't know latex, you can how other people made pretty equations by clicking on them.
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