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.