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](https://mathoverflow.net/questions/14456/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 $\infty$. 



Sums of squares in global fields (Reference Request)
Corollary 1.5 on page 379 of Lam's *Introduction to quadratic forms over fields* (in the Sums of Squares section) states that if $F$ is any nonreal global field then every element of $F$ is a sum of four squares. Since you say you are interested in finite extensions of $\mathbb F\_q(t)$, and all fields of characteristic other than $0$ are necessarily nonreal, it follows that the answer to your question "yes".


