TITLE: Why sum of three squares of real polynomials is a sum of two squares?
QUESTION [15 upvotes]: If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are fields for which such a statement holds?

REPLY [9 votes]: Theorem 4.2 from this paper says that in general you will need the coefficients of the two squares to be algebraic numbers of exponential, in $\deg (f^2+g^2+1)$, degrees. More precisely, if the Galois group of $f^2+g^2+1\in\mathbb{Q}[x]$ is the symmetric group $S_{d}$, then you will need algebraic numbers of degree exponential in $d$.