Find $1/\alpha$ when basis is {1, $\alpha$, $\alpha^2$, $\alpha^3$} I am trying to calculate a sub field and in the process, I need to state $1/\alpha$ in terms of $\alpha$.
Now, my $\alpha$ = $\sqrt{3+\sqrt{20}}$.
I can't for the life of me do this simple calculation!!
The minimal polynomial is $x^4-6x^2-11$ and I know I should be able to state $1/\alpha$ in terms of 1, $\alpha$, $\alpha^2$ and $\alpha^3$ but it is just not working for me.
Can someone please explain to me how to get the solution??

Just rewrite the minimal equation as
$$\alpha^4-6\alpha^2=\alpha(\alpha^3-6\alpha)=11,$$
so $\quad\alpha^{-1}=\frac1{11}(\alpha^3-6\alpha)$.
