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 $K$.
$x^4-2=x^4+1=(x^2-x-1)(x^2+x-1)$
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 $|K|=3^2$.
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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
$x^8+1=x^8+4=(x^4-2x^2+2)(x^4+2x^2+2)$
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($3^4$), $|K|=3^4$
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Question1.
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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$\color{red}{Question2.}$
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).
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Thanks in advance.
And I apologize if a sentence cannot be smooth.