  Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field
$K \subseteq \mathbb C$. The $K$-rank of $f$ is the smallest number of $d$-th
powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We
prove that for $d \ge 5$, there always exists a form of degree $d$ with at
least three different ranks over various fields. The $K$-rank of a form $f$
(such as $x^3y^2$) may depend on whether -1 is a sum of two squares in $K$.
