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I was given a homework of number theory which consists in studying a splitting field of a particular polynomial: I won't ask directly about the problem, however there are many basic doubts about the general theory which bother me.

Suppose we have a number field of the form $\mathbb{F}=\mathbb{Q}[X]/(f(X))$, where $f(X)\in\mathbb{Z}(X)$ is a degree four irreducible monic polynomial, and suppose all roots are complex: then $$f(X)=(X-\beta_1)(X-\bar{\beta}_1)(X-\beta_2)(X-\bar{\beta}_2)$$ (Writing out explicitly the roots seems rather useless, as they look pretty awful, I checked numerically)

i) Knowing that $\mathbb{F}=\mathbb{Q}[\alpha]$, where $\alpha$ is the class of $X$ in the quotient, is there any way to write out explicitely the automorphisms $\operatorname{Hom}_\mathbb{Q}(\mathbb{F,C})$? My guess is that $\alpha$ should be sent to some $\beta_i$ or $\bar{\beta}_i$

ii) To compute the ring of integers $\mathcal{O}_\mathbb{F}$ we were indicated the notes by René Schoof (http://www.mat.uniroma2.it/~eal/moonen.pdf, appendix to chapter 4). Suppose that for example $2$ divides the discriminant $\Delta(1,\alpha,\alpha^2,\alpha^3)$, then the algorithm demands to check for integrality all linear combinations of the form $$\frac{n_0}{2}+\frac{n_1}{2}\alpha+\frac{n_1}{2}\alpha^2+\frac{n_1}{2}\alpha^3$$ where the $n_i\in\{0,1\}$. This seems a rather lengthy and random check: is there any way to optimize the procedure? For example, I thought about calculating the norm of these elements and then verify that it is in fact an integer, yet that would imply either knowing how the group $\operatorname{Hom}_\mathbb{Q}(\mathbb{F,C})=\{\sigma_1,...,\sigma_4\}$ behaves on the elements $\alpha^i$ and then use $Norm(x)=\Pi\sigma_i(x)$, or computing the characteristic polynomials of all such elements.

iii) When it comes to descrive $\mathcal{O}_\mathbb{F}^*$ I wish to use Dirichlet's Theorem: however, is there an optimal way to look for roots of unit? All examples given during the course (and in the notes) exploit the fact that the extension has at least one real embedding and thus $\{roots\ of\ unit\}=\{1,-1\}$, which is not my case

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