The number of non-square elements in a finite field $F$ of odd characteristic is $(|F| - 1)/2$, expressed as the cardinality of the filter of elements on which the quadratic character takes the value $-1$.
Injectivity of the Möbius-like map $a \mapsto (\alpha + a)(\beta + a)^{-1}$ on the locus where $\beta + a \neq 0$, provided $\alpha \neq \beta$.
Theorem 3.7 (Rabin). For distinct $\alpha, \beta$ in a finite field $F$ of odd characteristic, exactly $(|F| - 1)/2$ shifts $\delta$ satisfy $\alpha + \delta, \beta + \delta \neq 0$ and the quadratic characters of these shifts disagree.
Evaluating the diagonal ternary quadratic form $aX_0^2 + bX_1^2 + cX_2^2$ at a point $v$ yields $a v_0^2 + b v_1^2 + c v_2^2$.
Theorem 3.4. Every non-degenerate diagonal ternary conic $aX^2 + bY^2 + cZ^2 = 0$ over a finite field $K$ of odd characteristic has a nonzero rational point. This is a special case of the Chevalley–Warning theorem.
The number of affine solutions $(x, y, z) \in K^3$ to the diagonal ternary quadratic form $aX^2 + bY^2 + cZ^2 = 0$.
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The number of projective points on the conic $aX^2 + bY^2 + cZ^2 = 0$, obtained from the affine count by removing the origin and quotienting by the action of $K^\times$.
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If $-b/a$ is not a square in a field $K$, then the only solution to the binary form $a x^2 + b y^2 = 0$ is the trivial one $x = y = 0$.
The Jacobi-sum identity $\sum_{a \in K} \chi(a(a-1)) = -1$, where $\chi$ is the quadratic character of a finite field of odd characteristic.
For $d \neq 0$ in a finite field $K$ of odd characteristic, $\sum_{t \in K} \chi(t^2 + d) = -1$, where $\chi$ is the quadratic character.
For nonzero $e, c$ in a finite field of odd characteristic, $\sum_{z \in K} \chi(e + c z^2) = -\chi(c)$.
The number of solutions $x \in K$ to $ax^2 + d = 0$ in a finite field of odd characteristic equals $1 + \chi(-d/a)$, where $\chi$ is the quadratic character.
The double character sum $\sum_{y, z \in K} \chi(b y^2 + c z^2) = 0$ for nonzero $b, c$ in a finite field of odd characteristic.
Fibered description of solutions of $aX_0^2 + bX_1^2 + cX_2^2 = 0$ over the projection onto the last two coordinates: equivalent to a $\Sigma$-type indexed by $(y, z)$ of solutions to $aX^2 + (b y^2 + c z^2) = 0$.
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In characteristic 2, the number of affine solutions to a non-degenerate diagonal ternary quadratic form $aX^2 + bY^2 + cZ^2 = 0$ over a finite field $K$ equals $|K|^2$ (exploiting that Frobenius $x \mapsto x^2$ is bijective).
In odd characteristic, the number of affine solutions to a non-degenerate diagonal ternary quadratic form $aX^2 + bY^2 + cZ^2 = 0$ over a finite field $K$ equals $|K|^2$. Established via a character-sum argument.
Uniform statement (any characteristic): for a non-degenerate diagonal ternary quadratic form over a finite field $K$, the number of affine solutions is $|K|^2$.
In characteristic 2, the "double line" conic $aX^2 + bY^2 = 0$ also has $|K|^2$ affine solutions: since the form is a perfect square, every $(y, z)$ extends uniquely to a solution.
For the split conic $aX^2 + bY^2 = 0$ where $-b/a$ is a square in a finite field of odd characteristic (so the form factors over $K$), the number of affine solutions equals $2|K|^2 - |K|$.
A geometrically irreducible diagonal conic $aX^2 + bY^2 + cZ^2 = 0$ over $\mathbb{F}_q$ (with $abc \neq 0$) has exactly $q + 1$ projective points.
In characteristic 2, the degenerate "double-line" conic $aX^2 + bY^2 = 0$ over $\mathbb{F}_q$ has exactly $q + 1$ projective points.
A conic of the form $aX^2 + bY^2 = 0$ that splits over the base field ($-b/a$ a square in $K$) has $2q + 1$ projective points: it is a union of two lines meeting in one point.
A conic $aX^2 + bY^2 = 0$ that splits only over a quadratic extension ($-b/a$ not a square in $K$) has exactly $1$ projective $K$-point: the lone point $[0 : 0 : 1]$ visible to $K$.
A diagonal conic $aX^2 + bY^2 + cZ^2 = 0$ is geometrically irreducible when $c \neq 0$.
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A diagonal conic $aX^2 + bY^2 + cZ^2 = 0$ splits over the base field $K$ when $c = 0$ and $-b/a$ is a square in $K$.
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A diagonal conic $aX^2 + bY^2 + cZ^2 = 0$ splits only over a quadratic extension when $c = 0$ and $-b/a$ is not a square in $K$.
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Corollary 3.5 (trichotomy, odd characteristic). Over a finite field of odd characteristic, a diagonal conic $aX^2 + bY^2 + cZ^2 = 0$ (with $a, b \neq 0$) is either geometrically irreducible (with $q + 1$ points), split over $K$ (with $2q + 1$ points), or split only over a quadratic extension (with $1$ point).
Characteristic-agnostic trichotomy: over any finite field $K$, the number of projective points on a diagonal conic with $a, b \neq 0$ is either $q + 1$, $2q + 1$, or $1$.
The number of projective points on a diagonal conic over a finite field $\mathbb{F}_q$ is always congruent to $1$ modulo $q$.
Equivalent divisibility statement: $q = |K|$ divides $\#C(\mathbb{F}_q) - 1$ for any diagonal conic $C$ with $a, b \neq 0$.