How can we know the root of 8-th cylotomic polynomial $\mod 3$?

Question

Suppose we have 8-th cyclotomic polynomial $\Phi_8(x)=x^4+1$

It is standard fact that any cyclotomic polynomial is irreducible over rational field.

When the coefficient is over $\mathbb{Z}_3$ i.e., $\mod 3$, it can be factorized.

How can we factorize $\Phi_8(x)$?

I know that the number of factor is 2, but am not sure what the factors are.

Answer 1

Modulo $3,$ you have $x^4+1= \left(x^2+x+2\right) \left(x^2+2 x+2\right).$