I have an algebraic field $\mathbb Q(\gamma)$ with $\gamma$ the complex root of $X^3+X^2+X-1$, i.e., $\gamma\approx-0.771+1.115\mathrm i$.
I have two closely related questions:
Is $\mathbb Q(\gamma)$ closed under complex conjugation?
If so, how can one express $\overline\gamma$ as an element of $\mathbb Q(\gamma)$, i.e. as $\overline\gamma=a+b\gamma+c\gamma^2$ for $a,b,c\in\mathbb Q$?
And it is possible to answer this question more in general?