Math Help - Method for finding a Minimal Polynomial?

1. Method for finding a Minimal Polynomial?

Hello, I cannot seem to find in my notes nor in any text a method of finding minimal polynomials. Most minimal polynomials I've encountered are somewhat trivial, but I came across one that I may be tested on that I am unable to solve using a brute-force method:

$m_{\sqrt 2+i}$ over $\mathbb Q$

The reason I ask about a method of finding the min. poly. is because I am taking an oral test, and I don't want to be floundering about if I am asked something different than this one in particular.

Anyway, I would appreciate any help. Thanks!

P.s.: That's m_{\sqrt 2 +i}; sorry that the "i" is so tiny.

2. Originally Posted by Dark Sun
Hello, I cannot seem to find in my notes nor in any text a method of finding minimal polynomials. Most minimal polynomials I've encountered are somewhat trivial, but I came across one that I may be tested on that I am unable to solve using a brute-force method:

$m_{\sqrt 2+i}$ over $\mathbb Q$

The reason I ask about a method of finding the min. poly. is because I am taking an oral test, and I don't want to be floundering about if I am asked something different than this one in particular.

Anyway, I would appreciate any help. Thanks!

P.s.: That's m_{\sqrt 2 +i}; sorry that the "i" is so tiny.
$x=\sqrt{2}+i$

$x^2=(\sqrt{2}+i)^2 \iff x^2=2+2i\sqrt{2}-1 \iff x^2-1=2i\sqrt{2}$

$(x^2-1)^2=(2i\sqrt{2})^2$

$x^4-2x^2+1=-8$

$x^4-2x^2+9=0$

3. Originally Posted by TheEmptySet
$x=\sqrt{2}+i$

$x^2=(\sqrt{2}+i)^2 \iff x^2=2+2i\sqrt{2}-1 \iff x^2-1=2i\sqrt{2}$

$(x^2-1)^2=(2i\sqrt{2})^2$

$x^4-2x^2+1=-8$

$x^4-2x^2+9=0$
you also need to prove that $p(x)=x^4 - 2x^2 + 9$ is irreducible over $\mathbb{Q}$: we only need to show that it cannot be factored into two quadratic polynomials over $\mathbb{Z}.$ why?

suppose $p(x)=(x^2+ax+b)(x^2+cx+d),$ for some $a,b,c,d \in \mathbb{Z}.$ then we'll have $bd=9$ and $b+d+2=a^2,$ which is easily seen to have no solutions in $\mathbb{Z}.$

4. Because irreducible over $\mathbb Z$ implies irreducible over $\mathbb Q$.

$x^4-2x^2+9$ clearly does not have any degree 1 factors. Also, a degree 3 factor would imply a degree 1 factor, so if a factor exists, then it will be degree 2, which you have just shown is a contradiction.

Thank you EmptySet and NonCommAlg for presenting me with this truly powerful method!