# splitting field over $Z_3$ (for large degree of polynomial)

Could you verify(or advise) this solving process?

After I solve some typical exercise concerned with splitting field, Galois group,

I made a following problem. But 'large degree' of f(x) bother me...



The origin exercise. (I want some advice for my solving process.)

Let $K$ be a splitting field of a polynomial $x^4 -2$ over a field $Z_3$.

Find a order of $K$.

$x^4-2=x^4+1=(x^2-x-1)(x^2+x-1)$

and $x^2-x-1$ and $x^2+x-1$ are irreducible.

GF($3^2$)= $\{$roots of $x^{{3}^2}-x$$\} And (x^2-x-1)(x^2+x-1) divides x^{{3}^2}-x Therefore K=GF(3^2). Hence |K|=3^2.$$ $$(And I made a problem like this) Let K be a splitting field of a polynomial x^8+1 over a field Z_3 What is order of K (i.e. |K|) I used Germain identity, so I get x^8+1=x^8+4=(x^4-2x^2+2)(x^4+2x^2+2) and x^4-2x^2+2 and x^4+2x^2+2 are irreducible. GF(3^4)=\{ roots of x^{3^4} -x \} And (x^4-2x^2+2)(x^4+2x^2+2) divides x^{3^4} -x Therefore K=GF(3^4), |K|=3^4$$$$Question1. Is my process right? Could you give me some advice, please? (Actually I think ... I have a superficial knowledge about splitting field)$$ $$\color{red}{Question2.} In the second problem (I made), degree 8 is large for me. If I didn't come up with Germain identity,,,, How can I solve this problem?? Id est, I'd like to know '\color{red}{the} \color{red}{general} \color{red}{method}' of solving these type (large degree).$$$\$