For which n the equation $x^4+x^3+x^2+x+1=0$ has roots in the field of $3^n$ elements?
I don't have any ideas of how the equation's roots depend on the field's elements. Does the number of elements $3^n$ means that we consider the equation modulo $3^n$?
No, it means the "Galois field" of order $3^n$. This is isomorphic to $\Bbb F_3[X]/\left<f(x)\right>$ where $f$ is any irreducible degree $n$ polynomial over $\Bbb F_3$. Anyway the roots of $x^4+x^3+x^2+x+1=0$ are the primitive fifth roots of unity. A field of order $3^n$ has primitive fifth roots of unity iff $5\mid(3^n-1)$. This is because $3^n-1$ is the order of the multiplicative group of the field.