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? 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 K and ab , 0 are in K. So a ~ b for every a, b in K \ {0}.

However the author proceeds: Does the analogous statement hold if one replaces 2 by some arbitrary power 2^n? (supposedly the first statement was ~ is an equivalence relation). And here I could write ab= ab + 0 + 0 + 0 (for n= 2). But this seems too obvious. There must be something I don't understand well in the statement of the problem.

Any hint will be welcome. Regards.