I need an example of the use of Euler's Theorem for polynomials

Question

I need an example of the use of this theorem for $\mathbb Z / 3$ if possible.

Let ∈ [], a finite field and: ()={ ∈ []: 0 ≤ ≤ and (,)=1} So, for all ∈ [] with (,)=1, we have: ^(())≡ 1 ( )

For instance, apply this to $x^3 + 2x + 1$

PS: Consider as the degree of a polynomial

Answer 1

Perhaps this is a bit un-imaginative, but we can perform pseudo-primality tests:

Claim: $m=x^3+x+1$ is not irreducible in $\mathbb{Z}_3[x]$.

Proof: If not, then $\varphi(m)=\left|\mathbb{Z}_3\right|^{\deg m}-1=26$. Thus by Euler's theorem, $x^{26}\equiv 1\pmod{m}$. However by explicit division with remainder, we find that $$x^{26}\bmod m=x^2\ne 1$$ Thus $m$ is not irreducible. Indeed, $m=(x-1)(x^2+x-1)$.