Given an irreducible polynomial $p$ of degree $n$ in $\mathbb{R}[X]$, does there exists an $n\times n $ real matrix with $p$ as its characteristics polynomial.
Since similar matrices have the same characteristic polynomial. The set $P_p:=\{A \in M_n(\mathbb{R})\ | \chi_{A}=p\}$ is closed under conjugation. Then P can be written as a union of disjoint conjugacy classes (of similar matrices). The question is how many conjugacy classes?
I have a partial answer to this question. If $\ p=x^2+x+1$ then their are $2\times 2$ matrices $A$ with characteristic polynomial $p$, where $tr(A)=-1\ $ and $\ det(A)=1$ so $$A=\left( \begin{matrix} -1 & 1 \\ -1 & 0 \end{matrix} \right)$$ has $\chi_A= x^2+x+1$. But I do not know how many conjugacy classes is the set $P_{x^2+x+1}$ composed of?