A different example: minimal polynomial of $\displaystyle u=\sqrt2+\sqrt3$. This belongs to $\displaystyle [\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]$ and basis for $\displaystyle \mathbb{Q}(\sqrt2,\sqrt3)$ is $\displaystyle 1,\sqrt2,\sqrt3,\sqrt6$. (You might try to prove this from the theory you've already learned.) Now:

$\displaystyle (\sqrt2+\sqrt3)^0=1$

$\displaystyle (\sqrt2+\sqrt3)^1=\sqrt2+\sqrt3$

$\displaystyle (\sqrt2+\sqrt3)^2=5+2\sqrt6$

$\displaystyle (\sqrt2+\sqrt3)^3=11\sqrt2+9\sqrt3$

$\displaystyle (\sqrt2+\sqrt3)^4=49+20\sqrt6$

So you just need to find the coefficients by which you obtain zero as a linear combination of the vectors $\displaystyle (1,0,0,0)$, $\displaystyle (0,1,1,0)$, $\displaystyle (5,0,0,2)$, $\displaystyle (0,11,9,0)$, $\displaystyle (49,0,0,20)$.

You should finally arrive to the minimal polynomial $\displaystyle x^4-10x^2+1$.