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Math Help - Equivalence relation over a field

  1. #1
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    Equivalence relation over a field

    Hi:
    Let K be a field and, for a, b in K \ {0}, write a ~ b if ab is a sum of two squares in K. The author writes: why is ~ an equivalence relation?

    One proof for transitivity is
    ab = x^2 + y^2
    bc = z^2 + w^2
    acb^2 = (xz)^2 + (xw)^2 + (yz)^2 + (yw)^2 =
    = [(xz + yw)^2 + (xw - yz)^2].

    However, I see that for any a, b in K \ {0}, ab= ab + 0, where 0 is a square and ab is the square of an element in some extension of E of K and ab, 0 are in K. So a ~ b for every a, b in K \ {0}. That such an E exists amounts to saying that the polynomial x^2 - ab belonging to K[x] has a root in some extension of K, which is certainly true. What's wrong with this line of reasoning, if wrong at all?

    Any hint will be welcome. Regards.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    The relation is on K. It is not on the algebraic closure of K, because as you noticed it would make the problem trivial.

    Recall that a relation on a set X is a subset of X \times X. An equivalence relation R is a relation such that

    (1) (x,y) \in R \Rightarrow (y,x) \in R
    (2) (x,x) \in R \ \ \forall\ \  x \in K
    (3) \{(x,y),(y,z)\} \subset R \Rightarrow (x,z)\in R
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  3. #3
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    Quote Originally Posted by Bruno J. View Post
    The relation is on K. It is not on the algebraic closure of K, because as you noticed it would make the problem trivial.
    Then the statement should be "Let K be a field and, for a, b in K \ {0}, write a ~ b if ab is the sum of two squares OF ELEMENTS in K". This my humble opinion. Thanks for your reply. Adieu.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    I quote your original question :

    Let K be a field and, for a, b in K \ {0}, write a ~ b if ab is a sum of two squares in K. The author writes: why is ~ an equivalence relation?
    There is no ambiguity whatsoever.
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  5. #5
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    Quote Originally Posted by Bruno J. View Post
    I quote your original question :



    There is no ambiguity whatsoever.
    Then I'm a null entity at mathematics. Don't think I have comfortably recourse to MHF to solve the problem. I've tried hard at understanding the statement of the problem but it makes no sense to me. "ab is a sum of two squares in K": what does this mean? For me it is: "there exist x, y such that ab = x^2 + y^2, with x^2, y^2 in K". I understand the last statement doesn't make much sense, because x and y must exist somewhere. It is because of that I am forced to assumed an extension where to place x and y. So the last statement is left now in this form:
    "there exist x,y in some extension of K such that ab = x^2 + y^2, with x^2 ,y^2 in K". But this makes the problem trivial. So, I do not see a way out.

    Grateful for your reply.
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