Splitting field and degree of extension of $x^4 + 2x^2 -2$ over $\mathbb Q$ and $\mathbb F_{13}$
I already solved it over $\mathbb Q$, but I don't know how it's done over finite fields. I tried looking up some examples but didn't really get what they are doing Could you show me how?. Is it a good idea to do $y=x^2$?
As for the degree of the extension all I know is that it must divide 4!, because there's a theorem: the degree of the extension of the splitting field of a polynomial of degree n divides n!
Best Answer
Try it for $x^2 + 1$ and $x^2+5$ for better understanding.
As the comments suggested, $f=(x-4)(x+4)(x^2+5)$. So to get the splitting field, you only have to add the roots of $x^2+5=0$ which would be a degree 2 extension.