For cardinality reasons, we know that every element in a finite field $F$ is a sum of two squares. If I fix $a,b\in F$ with $a,b\neq 0$, can every element in $F$ be written in the form $aX^2 + bY^2$ for some $X,Y$ in $F$? If not, can we say how many solutions there are in terms of $a$ and $b$?
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4Exactly the same cardinality reasons apply, since $x \mapsto ax$ is injective. – Erick Wong May 17 '13 at 19:47