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Math Help - Jacobi Symbol Properties

  1. #1
    Senior Member
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    Arrow Jacobi Symbol Properties

    This is a theorem about Jacobi symbols in my textbook:
    Let n and m be ODD and positive. Then (a/nm)=(a/n)(a/m) and (ab/n)=(a/n)(b/n)
    Moreover,
    (i) If gcd(a,n)=1, then ( a^2/n) = 1 = ( a/n^2)
    (ii) If gcd(ab,nm)=1, then ( ab^2/nm^2)=(a/n)
    =====================================

    (i) is easy and follows from the definition, but how can we prove (ii)? My textbook stated the theorem without proof and just says the proofs are easy, but I have no idea why (ii) is true.

    Any help is appreciated!
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  2. #2
    Super Member Deadstar's Avatar
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    Quote Originally Posted by kingwinner View Post
    This is a theorem about Jacobi symbols in my textbook:
    Let n and m be ODD and positive. Then (a/nm)=(a/n)(a/m) and (ab/n)=(a/n)(b/n)
    Moreover,
    (i) If gcd(a,n)=1, then ( a^2/n) = 1 = ( a/n^2)
    (ii) If gcd(ab,nm)=1, then ( ab^2/nm^2)=(a/n)
    =====================================

    (i) is easy and follows from the definition, but how can we prove (ii)? My textbook stated the theorem without proof and just says the proofs are easy, but I have no idea why (ii) is true.

    Any help is appreciated!
    Think it is because...

    \bigg{(}\frac{ab^2}{nm^2}\bigg{)} = \bigg{(}\frac{a}{nm^2}\bigg{)}\bigg{(}\frac{b^2}{n  m^2}\bigg{)} = \bigg{(}\frac{a}{nm^2}\bigg{)}\cdot 1

    = \bigg{(}\frac{a}{n}\bigg{)}\bigg{(}\frac{a}{m^2}\b  igg{)} = \bigg{(}\frac{a}{n}\bigg{)}\cdot 1 = \bigg{(}\frac{a}{n}\bigg{)}

    Was it supposed to be \bigg{(}\frac{(ab)^2}{(nm)^2}\bigg{)}? That was a bit unclear
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  3. #3
    Senior Member
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    Smile

    Quote Originally Posted by Deadstar View Post
    Think it is because...

    \bigg{(}\frac{ab^2}{nm^2}\bigg{)} = \bigg{(}\frac{a}{nm^2}\bigg{)}\bigg{(}\frac{b^2}{n  m^2}\bigg{)} = \bigg{(}\frac{a}{nm^2}\bigg{)}\cdot 1

    = \bigg{(}\frac{a}{n}\bigg{)}\bigg{(}\frac{a}{m^2}\b  igg{)} = \bigg{(}\frac{a}{n}\bigg{)}\cdot 1 = \bigg{(}\frac{a}{n}\bigg{)}
    Why (b^2/nm^2)=1? How do you know that gcd(b,nm^2)=1?
    Why (a/m^2)=1? How do you know that gcd(a,m)=1?

    Was it supposed to be \bigg{(}\frac{(ab)^2}{(nm)^2}\bigg{)}?
    No.

    Thanks for explaining!
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by kingwinner View Post
    Why (b^2/nm^2)=1? How do you know that gcd(b,nm^2)=1?
    Why (a/m^2)=1? How do you know that gcd(a,m)=1?
    Since  \left(\frac{xy}{r}\right) = \left(\frac{x}{r}\right)\left(\frac{y}{r}\right) , we have that  \left(\frac{b^2}{nm^2}\right) = \left(\frac{b}{nm^2}\right)\left(\frac{b}{nm^2}\ri  ght) = \left(\frac{b}{nm^2}\right)^2 =1

    The other case follows similarly.
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