Find all zeroes $f(x) = 3x^{299}+2x^{222}-5 \in \mathbb{F}_{7}[x]$ in $\Bbb{F}_{7}$. I know that we can just test out all the values, because the field is small. But I'm interested. Is there a more optimal algorithm to solve that problem?
2026-04-25 00:15:57.1777076157
Find all zeroes $f(x) = 3x^{299}+2x^{222}-5 \in \mathbb{F}_{7}[x]$ in $\Bbb{F}_{7}$
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Since Clearly $\bar{0}$ is not the solution. So the solutions are from $\mathbb{F}_7^*$. Since the order is $6$. We can reduce $x^{299} = x^5$ and $x^{222} = \bar{1}$ so the we have just to solve for $3x^5 +2-5 = 3x^5-3$.
So we have to solve for $x^5-1$ which is easy to solve. The answer is $1$ clearly because $5$ is coprime to the order of the group $\mathbb{F}_7^*$ which is $6$.