Looking for the splitting field of polynomials over $\mathbb{F}_4$

Question

How to find the degrees of splitting fields of $x^7-1$ and $x^8-1$ over $\mathbb{F}_4$?

For $x^8-1$ I wrote $$ x^8-1=(x^2+i)(x+ \sqrt{i})(x- \sqrt{i})(x+i)(x-i)(x+1)(x-1), $$ but I'm not sure what to do with $(x^2+i)$... Is there a brutal force for this kind of problem?

Answer 1

In $\Bbb F_4$, $1+1=2=0$ and $-1=1-2=1$. Therefore $$(x-1)^2=x^2-2x+1=x^2-1.$$ So, $$(x-1)^4=(x^2-1)^2=x^4-1$$ and $$(x-1)^8=(x^4-1)^2=x^8-1.$$