I have to see whether $x^2+x-1\in\mathbb{F}_3[x]$ is irreductible. One first way to see this is by checking if there exist any roots of the polynomial on the field, the problem is that I don't understand which field they are talking about; $\mathbb{F}_3[x]=\{0,1,x\}$? Or is it $\mathbb{Z}_3$? In such case there are no roots since substituing each element into the polynomial yields a result which is not congruent with $0$ (mod $3$).
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1${0,1,x}$ is not a field, it doesn't contain $-x$ or $1+x$ or $1/x$ for instance. The field is $\Bbb F_3$, of course. – coiso Jun 01 '23 at 06:34
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3Also $\Bbb F_3[x]$ is not the set ${0,1,x}$. It has infinitely many elements. – pancini Jun 01 '23 at 06:35
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You can check out this post https://math.stackexchange.com/q/1343450/1093844 – Soumik Mukherjee Jun 01 '23 at 06:44
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Also check this out if you are confused about the structure of a finite field $\Bbb{F_{p^n}}$ – Soumik Mukherjee Jun 01 '23 at 06:51
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Let $f\in \Bbb K[X]$, with $\Bbb K$ field and $\deg f=2$ or $\deg f=3$.
Then $f$ is irreducible in $\Bbb K[X]$ iff $f$ has no roots in $\Bbb K$.
In this case, $\Bbb K= \Bbb F_3$, so you only have to check whether $0$, $1$ or $2$ are roots of $f$ to prove $f$ is irreducible.
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