Why $s(1-s)$ numbers which are squares in a field are written $\frac{u^2}{1+u^2}$? Trying to exercise in Math again after years of other activities, I need a little help on this :
Let $F$ be some arbitrary field with characteristic > 2.
Let $S \subseteq F$ be the set of numbers s such that $s(1-s)=a^2$ for some $a \in F$ , and $s \neq 1$
Let $U \subseteq F$ be the set of numbers of the form $\frac{u^2}{1+u^2}$ for some $u \in F$, and $u^2 \neq -1$
We must show that the $S = U$.
I can easily show that $U \subseteq S$
ut I'm stuck on showing that $S \subseteq U$

Because $\frac{s}{1-s}$ is also a square.
