2
$\begingroup$

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).

$$ $$

Thanks in advance.

And I apologize if a sentence cannot be smooth.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.