Hi
I am wondering how to find the minimal polynomial of 2+\sqrt{i} over \mathbb{Q}.
I have got it to x^{4}-2x^{2}+9 but I don't know how to shaw that this is irreducible!
Thanks
I don't know from where did you get that polynomial. I got:
$\displaystyle x=2+\sqrt{i}\Longrightarrow (x-2)^2=i\Longrightarrow (x-2)^4=-1\Longrightarrow x^4-8x^3+24x^2-32x+17$ , and this polynomial is irreducible by Eisenstein's criterium with $\displaystyle p=2$ after carrying on the transformation $\displaystyle x \rightarrow x+1$
Tonio