Show that the complex numbers $i\sqrt{3}$ and $1+i\sqrt{3}$ are roots of $f(x) = x^{4}-2x^{3} + 7x^{2} - 6x + 12$. Let $K$ be the field generated by $\mathbb{Q}$ and the roots of $f$. Is there an automorphism $\sigma$ of $K$ with $\sigma(i\sqrt{3}) = 1 + \sqrt{3}$?
The first part is easy. For the second, I tried calculate $$[\mathbb{Q}(i\sqrt{3},-i\sqrt{3},1 + i\sqrt{3},1-i\sqrt{3}):\mathbb{Q}]$$ but I couldn't advance. I don't know if this is the best way. Any hint?
Update. In fact, automorphism does not have to fix $\mathbb{Q}$, right?