Let $\mathbb{F}_q$ be a finite field of $q$ elements such that $q=p^{t}$. Let $\alpha$ be a solution of the equation $Ax^2 +Bx +C=0$ and $\alpha'$ its conjugate, where $A$, $B$ and $C$ are nonzero polynomials of $\mathbb{F}_q[T]$. What is the relation between $\alpha$ and $\alpha'$? what are the known properties of the two (existence, equality, absolute value, ...). Is there any accessible summary or reference that contains some answers?
2026-04-02 20:17:09.1775161029
A quadratic solution and its conjugate
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