I was trying to prove that for a given $n$, there exist a totally real number field of degree $n$.
I understood that it was equivalent to find a polynomial $P$ such that
(i) $P\in\mathbb Q[X]$ ;
(ii) $P$ is irreducible ;
(iii) $P$ only has real roots in $\mathbb C$ ;
(iv) $P$ has degree $n$.
But the construction is abstract, and I can not deduce from it an explicit polynomial satisfying the four conditions.
For a given $n$, do you know how can I construct an explicit polynomial $P$ satisfying (i), (ii), (iii) and (iv)?