Sums of squares are closed under division Surprisingly, we got only one question for our 2-hour exam and I think nobody solved it. Here is the problem:

Assuming that $K$ is a field, show that $S$ is stable under addition,  multiplication and division, where $S$ is defined as follow:
  $$S=\left\{\sum_{i=1}^{n}{x_i}^2 \mid n\in \Bbb N ,\ x_i\in K\right\}.$$


$$\frac{1}{x_1^2+x_2^2+x_3^2}=\frac{x_1^2}{(x_1^2+x_2^2+x_3^2)^2}+\frac{x_2^2}{(x_1^2+x_2^2+x_3^2)^2}+\frac{x_3^2}{(x_1^2+x_2^2+x_3^2)^2}$$
