Let $K$ be the splitting field of $X ^ { n } - 1$ over $\mathbb { F } _ { p }$, where $n$ is a positive integer. Pick the correct statement(s) from below. (A) $K$ has $p ^ { n }$ elements. (B) If $p \nmid n$, then the group of field automorphisms of $K$ is isomorphic to the multiplicative group $( \mathbb { Z } / n \mathbb { Z } ) ^ { \times }$. (C) $K$ is a separable extension of $\mathbb { F } _ { p }$. (D) There exists $m > n$ such that $K$ is the splitting field of $X ^ { m } - 1$ over $\mathbb { F } _ { p }$.
Let $K$ be the splitting field of $X ^ { n } - 1$ over $\mathbb { F } _ { p }$, where $n$ is a positive integer. Pick the correct statement(s) from below.\\
(A) $K$ has $p ^ { n }$ elements.\\
(B) If $p \nmid n$, then the group of field automorphisms of $K$ is isomorphic to the multiplicative group $( \mathbb { Z } / n \mathbb { Z } ) ^ { \times }$.\\
(C) $K$ is a separable extension of $\mathbb { F } _ { p }$.\\
(D) There exists $m > n$ such that $K$ is the splitting field of $X ^ { m } - 1$ over $\mathbb { F } _ { p }$.