I have a question regarding the number of solutions of a equation over a finite field $\mathbb{F}_p$. First of all, consider the equation $x^3=a$ over $\mathbb{F}_p$, where $p$ is a prime such that $p\equiv 2 (\text{mod }3)$. The book that I'm currently reading says that this equation has exactly one solution in $\mathbb{F}_p$ for every $a\in \mathbb{F}_p$, because $\gcd(3,p-1)=1$, but the book does not prove this. Unfortunately, this doesn't convince me enough. Is there is a convincing elementary straightforward proof justifying why is this true?
Asked
Active
Viewed 61 times
1
-
This is a recurring question. It also appears often in the context of elliptic curves when tallying the solutions of an equation like $y^2=x^3+a$ over $\Bbb{F}_p$. Finding duplicate targets was not too taxing IMO. – Jyrki Lahtonen Jul 30 '21 at 06:56