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Math Help - alpha is a square of an element in Fq star

  1. #1
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    alpha is a square of an element in Fq star

    if q be a power of an odd prime. we need to prove that an element alpha belongs to the multiplicative group Fq star, is a square element of Fq star if and only if alpha ^(q-1)/2
    =1 . in particular -1 belongs to Fq star square if and only if q is congreunt to 1 (mod4).
    (Fq star is the multiplicative group that has size q-1).
    Thanks
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  2. #2
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    Quote Originally Posted by Mike12 View Post
    if q be a power of an odd prime. we need to prove that an element alpha belongs to the multiplicative group Fq star, is a square element of Fq star if and only if alpha ^(q-1)/2
    =1 . in particular -1 belongs to Fq star square if and only if q is congreunt to 1 (mod4).
    (Fq star is the multiplicative group that has size q-1).
    Thanks


    One direction is pretty easy: \displaystyle{\alpha=x^2\!\!\pmod q\Longrightarrow \alpha^{\frac{q-1}{2}}=x^{q-1}=1\!\!\pmod q\,\,\forall x\in\mathbb{F}_q^*}

    Now the other one: you'll need the following facts

    i) All the quadratic residues modulo q are \displaystyle{1^2\,,\,2^2\,,\,\ldots,\,\left(\frac  {q-1}{2}\right)^2} , and thus there are exactly

    \displaystyle{\frac{q-1}{2}} non-zero residues modulo q.

    ii) The polynomial \displaystyle{f(x):=x^{\frac{q-1}{2}}-1} has at most \displaystyle{\frac{q-1}{2}} different solutions in \mathbb{F}_q^*

    iii) Since all the quadratic residues modulo q are roots of f(x) above, no

    non-quadratic residue modulo q can be a root.

    iv)... and Q.E.D.

    Tonio
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  3. #3
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    Thanks Tonio, can you please tell me what does Q.E.D mean
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  4. #4
    Senior Member Shanks's Avatar
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    Q.E.D means quite elegantly done, that is to say, the solution ends here.
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  5. #5
    Senior Member roninpro's Avatar
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    Quote Originally Posted by Shanks View Post
    Q.E.D means quite elegantly done, that is to say, the solution ends here.
    Actually: Q.E.D. - Wikipedia, the free encyclopedia.
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