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

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- Feb 3rd 2010, 07:39 AMKirstyMinimal 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 - Feb 3rd 2010, 09:39 AMtonio

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 - Feb 3rd 2010, 10:11 AMKirsty
Oops,

I actually meant \sqrt{2}+i.

Sorry