An irreducible minimal polynomial of a real and complex matrix

Question

Let $A\in M_{n}(\mathbb{F})$ be a triangularizable matrix with complex spectrum where $ \mathbb{F} $ is real numbers set. What is a form of minimal polynomial of $A$ that is irreducible?

Now, if $\mathbb{F} $ is complex numbers set. What is a form of minimal polynomial of $A$ that is irreducible?

Answer 1

Case 1; $F=\mathbb{R}$. Then $A$ is a scalar matrix $A=aI_n$ (where $a\in\mathbb{R}$) or the minimal polynomial of $A$ is in the form $x^2+ax+b$, where $a^2-4b<0$, with roots $u\pm iv$ where $u\in\mathbb{R},v\in\mathbb{R}^*$. In the last case, $n=2p$ is even and $A$ is similar (over $\mathbb{R}$) to the matrix $diag(Z_1,\cdots,Z_p)$ where $Z_j=\begin{pmatrix}u&-v\\v&u\end{pmatrix}$.

Case 2. $F=\mathbb{C}$. Then $A$ is a scalar matrix $A=aI_n$ (where $a\in\mathbb{C}$).