Can sums of squares in a field always be written as the sum of four squares? Let $\mathbb{F}$ be a field. Let $S$ be the set of elements of $\mathbb{F}$ that can be written as a sum of squares. For $\mathbb{F} = \mathbb{Q},$ four squares might be required. Is there any field such that $S$ contains elements that aren't the sum of four squares? What is the smallest $n$ such that it is guaranteed that all elements of $S$ are the sum of $n$ squares?

For a field $F$, the minimal number $n$ such that every sum of squares in a $F$ is a sum of at most $n$ squares is called the Pythagoras number of $F$.
The discussion on MO here indicates that the calculation of the Pythagoras number for the rational function field $\mathbf R(x_1,\ldots,x_m)$ is still largely open.  It is known to be in the interval $[m+1,2^m]$, so for $m = 1$ the value is $2$. For $m = 2$, it is a theorem of Cassels, Ellison, and Pfister that the value is $4$. For $m \geq 3$ the exact value has not been determined.
Hoffmann, in 1998 here, showed every positive integer is the Pythagoras number of some formally real field (that means a field that is an ordered field in at least one way, or equivalently $-1$ is not a sum of squares in the field).
