# Minimal Polynomials

• February 3rd 2010, 07:39 AM
Kirsty
Minimal Polynomials
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
• February 3rd 2010, 09:39 AM
tonio
Quote:

Originally Posted by Kirsty
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:

$x=2+\sqrt{i}\Longrightarrow (x-2)^2=i\Longrightarrow (x-2)^4=-1\Longrightarrow x^4-8x^3+24x^2-32x+17$ , and this polynomial is irreducible by Eisenstein's criterium with $p=2$ after carrying on the transformation $x \rightarrow x+1$

Tonio
• February 3rd 2010, 10:11 AM
Kirsty
Oops,

I actually meant \sqrt{2}+i.

Sorry