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...

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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 $K$.


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 $|K|=3^2$.

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(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


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($3^4$), $|K|=3^4$



Is my process right?

Could you give me some advice, please?

(Actually I think ... I have a superficial knowledge about splitting field)

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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).

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Thanks in advance.

And I apologize if a sentence cannot be smooth.


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