I am currently reading notes on abstract algebra in which the following theorem is stated:
THEOREM : Let $\mathbb{F}$ be a field and consider the action of $GL_n(\mathbb{F})$ on $M_n(\mathbb{F})$ given by $$ m\mapsto gmg^{-1} $$ The orbits of ${GL}_n(\mathbb{F})$ are in bijection with block matrices $$ \begin{bmatrix} C_1 & & \\ & \ddots &\\ & & C_s \end{bmatrix} $$ where the sum of the sizes of the matrices $C_i$ is $n$, each $C_i$ is a matrix of the form $$ \begin{bmatrix} 0 & 0 &\dots & 0 & \ast \\ 1 & 0 & & 0 & \ast\\ & 1 & & \vdots & \vdots \\ & & & 1 & \ast \end{bmatrix} $$ and $\Delta(C_1)\mid \dots \Delta(C_s)$.
QUESTION 1: What does the author mean by "The orbits of ${GL}_n(\mathbb{F})$ are in bijection with block matrices"?
Furthermore, the theorem was followed by a corollary:
COROLLARY : If $A$ is a matrix in $M_n(\mathbb{F})$ and $\mathbb{K}$ is a field extension of $\mathbb{F}$ then the rational canonical form of $A$ over $\mathbb{K}$ is the same as over $\mathbb{F}$. Consequently, if $A_1, A_2$ are matrices in $M_n(\mathbb{F})$ and for some $B \in GL_n(\mathbb{K})$ we have $BA_1B^{-1} = A_2$ then for some matrix $\tilde{B}\in GL_n(\mathbb{F})$ we have $\tilde{B}A_1\tilde{B}^{-1} = A_2$.
QUESTION 2: How does this corollary follow from the theorem?
(1) An orbit can be regarded as "an similarity equivilent class of matrix". To be specific, 2 matrices are similar iff they are in the same class. So, the author means in each orbits there is a canonical one(the rational canonical form). And the bijection goes like following:
Let $A$ be the set of all canonical form, and $B$ be the set of all orbits. Then, there is a bijection between $A$ and $B$ in the following way: $f: A \to B, f(M) = O_M$,here $O_M$ stands for "the orbit of $M$, and $f^{-1}:B \to A$,is $f^{-1}(O) = R_O$, here $R_O$ stands for the (unique) rational form in $O$
(2) This can be stated as if two matrix are similar in a big field, so are them in smaller field. And, use the theory of rational canonical form, things are easy, because we know that 2 matrices are similar iff the have the same rational canonical form, so if they are similar in big field, their rational canonical forms are same, so do they in smaller field. So, they are similar in smaller field.