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Math Help - Finite field

  1. #16
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    incidentally all of the above prove that any element can be written both as x^2+y^2 and x^2-y^2
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  2. #17
    ynj
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    Thank you very much for the long discussion and the good proof!
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  3. #18
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    you are most welcome. i liked interacting with you. thank you too !!
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  4. #19
    ynj
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    Quote Originally Posted by nirax View Post
    call A as the multiplicative group of squares. so we decompose F = A \cup \{0\} \cup B. B is the set of non squares. suppose an element a is not expressible as sum of two squares. consider a - A. this set must be the same as B. consider x = a - a^2. clearly x \in B

    but B was the coset of the multiplicative group A in F^*. so B = a.A. it follows that there exists a y s.t

    x = a.y^2 = a - a^2

    \Rightarrow a (y^2 - 1 + a) = 0
    \Rightarrow y^2 = a - 1

    RHS belongs to B whereas LHS belongs to A which are disjoint.
    hmm ....shouldn't it be y^2=1-a???
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