I'm having some difficulty in finding the degree of the splitting field of a polynomial over a finite field. In particular $f = x^3 – 2$ over $\mathbb{F}_5$.
This polynomial factorises as $f(x) = (x-3)(x^2 + 3x + 4)$ over this field. I also know that the degree of the splitting field must at most $3! = 6$. Now I want to say that the extension field is $\mathbb{F}_5[x] / (x^2 + 3x + 4)$, in which case the extension would be degree 2, but how do I know that all the roots of $x^2 + 3x + 4$ are in this extension field?
Thanks
Best Answer
Call $K= \mathbb{F}_5[x]/(x^2+3x+4)$. Then the polynomial $x^2+3x+4 \in \mathbb{F}_5[x]$ has a root $ \alpha \in K$. But now, one can write $$x^2+3x+4 = (x- \alpha)g(x)$$ for some $g \in K[x]$. Since the degree of $x^2+3x+4$ is 2 and the degree of $x- \alpha$ is 1, $g$ must be a polynomial of degree 1, so it has a root $\beta \in K$.
Now, $x^2+3x+4$ can have at most two roots, so $\alpha, \beta \in K$ are all of them.