There is a result due to Siegel that, for a number field $K$, any totally positive element of $K$ is the sum of four squares of $K$. This is discussed in another question (sum of squares in ring of integers).

Is the result true in the more general setting of global fields? That is, for an algebraic function field $F$ is a totally positive element the sum of four squares?

Of course, in the number field terminology *totally positive* requires embeddings of $K$ into $\mathbb{R}$, which we do not have in the function field setting. So (I believe) we say an element $f \in F$ is *totally positive* if $f$ is a square in $F_{\infty}$, the completion of $F$ at the nonarchimedian infinite place $\infty$.