Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if the rational functions could have coefficients over Q rather than over R. Here is the relavant part of his speech
"At the same time it is desirable, for certain questions as to the possibility of certain geometrical constructions, to know whether the coefficients of the forms to be used in the expression may always be taken from the realm of rationality given by the coefficients of the form represented."
Does anyone know what these "certain geometrical constructions" are?
It seems maybe to me that Hilbert was attempting to embed rational projective space into higher dimensional rational projective space via these polynomials. Briefly, given a nonnegative homogeneous function $f(x_0,\ldots, x_n)$ with rational coefficients induces a metric on $QP^n$. Suppose Hilbert's dream holds that $f=p_0^2+\cdots + p_N^2$ where $p_i$'s are polynomials with rational coefficients. Then the map $p: QP^n\to QP^N$ where $p(x)=(p_0(x),\ldots, p_N(x))$ is an isometric embedding (almost!) where the metric induced by $f$ is the pullback back of the Euclidean metric on $QP^N$.
The above is just my hazard. But I would be delighted if anyone is aware of what exactly Hilbert's intended "geometrical constructions" are.