Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb { F } _ { p }$. Let $T : V \rightarrow V$ be a linear transformation. Select all the true statement(s) from below. (A) $T$ has an eigenvalue in $\mathbb { F } _ { p }$. (B) If $T ^ { p - 1 } = I$, then the minimal polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$. (C) If $T \neq I$ and $T ^ { p - 1 } = I$, then the characteristic polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$. (D) If $T ^ { p - 1 } = I$, then $T$ is diagonalizable over $\mathbb { F } _ { p }$.
Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb { F } _ { p }$. Let $T : V \rightarrow V$ be a linear transformation. Select all the true statement(s) from below.\\
(A) $T$ has an eigenvalue in $\mathbb { F } _ { p }$.\\
(B) If $T ^ { p - 1 } = I$, then the minimal polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.\\
(C) If $T \neq I$ and $T ^ { p - 1 } = I$, then the characteristic polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.\\
(D) If $T ^ { p - 1 } = I$, then $T$ is diagonalizable over $\mathbb { F } _ { p }$.