How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$.

Question

I have two questions:

1) How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$. Clearly the irreducible factors will be of degree $2$. But can anyone calculate it for me?

2) I have proved that if $f,g \in F[x]$ are irreducible polynomials of the same degree in a finite field $F$, then they have the same splitting field. Now using this how can I find the splitting field of $x^4 +1$?

How have you concluded that $x^4+1$ is not itself irreducible? — hmakholm left over Monica, Mar 10 '15 at 15:12
There are only 3 irreducible monic polynomials of degree 2, so trying all of them shouldn't take too long time. — hmakholm left over Monica, Mar 10 '15 at 15:19
x @user152715: There are exactly 9 polynomials of the form $x^2+bx+c$. Of these you can reject those with $c=0$ immediately. Futher, $(x-1)(x-2)$, $(x-1)^2$ and $(x-2)^2$ are not irreducible either. Then there are three left. — hmakholm left over Monica, Mar 10 '15 at 15:30
@user152715 How did you show the second part 2) if f,g are irreducible...? — user425181, Oct 23 '17 at 18:31

Abellan · Answer 1 · 2015-03-10T17:03:19.640

For the irreducibility part:

You can easily see that $x^4+1$ has no lineal factors so the only posibility will be two factors of degree 2.

Consider a monic polynomial in your field:

$f(x)=x^2+ax+b$

This polynomial is irreducible iff has no solutions in $\mathbb{F}_3$ so we have the following result:

$f$ is irreducible iff these conditions are satisfied

b$\neq$ $0$

$1+a+b\neq$ $0$

$4+2a+b\neq$ $0$

Now check if your polynial can be written as product of polynomials satisfying these conditions.

Every finite field extension is normal so if you add one root of your polynomial you'll get the splitting field

$\mathbb{F}_3$[$\alpha$] is a normal extension of $\mathbb{F}_3$ $\implies$ $\mathbb{F}_3$[$\alpha$] is the splitting field of the polynomial

$\mathbb{F}_3$[$\alpha$] is the splitting field – Abellan Mar 10 '15 at 17:32 — Abellan, Mar 10 '15 at 17:32

How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$.

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