Let $f$ be an irreducible cubic polynomial over $\mathbb{Q}$ with at most one real root and $\mathbb{k}$ the smallest subfield of $\mathbb{C}$ containing the roots of $f$. Choose the correct statement(s) from below:\\
(A) $\sigma(K) \subseteq K$ where $\sigma$ denotes complex conjugation;\\
(B) $[K : \mathbb{Q}]$ is an even number;\\
(C) $[(K \cap \mathbb{R}) : \mathbb{Q}]$ is an even number;\\
(D) $K$ is uncountable.