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Math Help - Here's the proof what's the question!

  1. #1
    Super Member Deadstar's Avatar
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    Here's the proof what's the question!

    This is a 4 part question on my tutorial and if you prove part d) the rest follow, except there's no part d) in the question! I can prove a), b) and c) without using whatever d) is but just thought I'd see if you guys know what this proof is actually proving!

    Q. Let p be an odd prime, and a, b be integers not divisible by p. Prove that
    a) a \equiv b (mod p) implies that \bigg{(}\frac{a}{p} \bigg{)} = \bigg{(}\frac{b}{p} \bigg{)}

    b) \bigg{(}\frac{ab}{p} \bigg{)} = \bigg{(}\frac{a}{p} \bigg{)}\bigg{(}\frac{a}{p} \bigg{)}.

    c) \bigg{(}\frac{a^2}{p} \bigg{)}=1; \bigg{(}\frac{a^2b}{p} \bigg{)} = \bigg{(}\frac{b}{p} \bigg{)}.

    (I though perhaps part d) was the second equality in part c) but I don't think the proof matches up.

    PROOF.

    a), b) and c) follow from d) so we should prove that first!
    d) Take a primitive root g, with a \equiv g^k mod p.

    Then a^{\tfrac{p-1}{2}} = g^{\tfrac{(p-1)k}{2}}.

    As g^{\tfrac{(p-1)}{2}} \equiv -1 mod p we see that...

    g^{\tfrac{(p-1)k}{2}} \equiv 1 mod p if k is even (i.e if a is a quadratic residue),

    while g^{\tfrac{(p-1)k}{2}} \equiv -1 mod p if k is odd (i.e a non quadratic residue). Hence result.


    I was thinking it was probably something like show that if g is a primitive root and k is even then,

    \bigg{(}\frac{g^k}{p} \bigg{)} = 1. and if k is odd then ... = -1.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Here's part a. The rest follow in a similar fashion.

     a\equiv b\bmod{p} \implies a^{(p-1)/2}\equiv b^{(p-1)/2}\bmod{p} \implies \left(\frac ap\right) = \left(\frac bp\right)
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  3. #3
    Super Member Deadstar's Avatar
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    Quote Originally Posted by chiph588@ View Post
    Here's part a. The rest follow in a similar fashion.

     a\equiv b\bmod{p} \implies a^{(p-1)/2}\equiv b^{(p-1)/2}\bmod{p} \implies \left(\frac ap\right) = \left(\frac bp\right)
    Lol I think you misread the question...

    I can prove a), b) and c) fine. My question was about this mystery part d) that only has a proof but the actual question was not on the tutorial.
    Basically, I've been given the proof, can you work out what the question was? My lecturers away so I can't ask him right now.

    I only included a,b and c so folk can see the whole question and what part d would imply.
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Deadstar View Post
    Lol I think you misread the question...

    I can prove a), b) and c) fine. My question was about this mystery part d) that only has a proof but the actual question was not on the tutorial.
    Basically, I've been given the proof, can you work out what the question was? My lecturers away so I can't ask him right now.

    I only included a,b and c so folk can see the whole question and what part d would imply.
    Probably too hard to deduce part d.
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  5. #5
    Super Member Deadstar's Avatar
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    Quote Originally Posted by chiph588@ View Post
    Probably too hard to deduce part d.
    Ah well thought I'd post it up to see...
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