Verifying equivalence relation

Question

Define $x$~$y$ if $xy$ are the sum of $2^n$ squares in a field $K$. Is this an equivalence relation? The only non-trivial thing to verify is transitivity, and I have no clues of whether it is true or not, let alone proving it or providing a counter-example. Could somebody give a hint? Thanks in advance.

P.S. I've verified it in the case of $n=1$ using complex number and $n=2$ using quaternion, but don't know how to continue. I think the difficulty is to define multiplication in $2^n$-dimension vector space over K.

Answer 1

Hint: You'll want to use Pfister's Theorem: In any field, the set of sums of $2^n$ squares is closed under multiplication.

Full Solution:

Then if $x\sim y$ and $y\sim z$, $xy$ and $yz$ are sums of $2^n$ squares, but then $xy^2z$ is a sum of $2^n$ squares by Pfister's theorem. Dividing by $y^2$, we see $xz$ is a sum of 2^n squares, so $x\sim z$.