Galois theory — an irreducible degree-p polynomial with two complex roots

Let $p\ge3$ be a prime number and $f(x)\in\mathbb{Q}[x]$ an irreducible polynomial of degree $p$. Suppose that $a_1,…,a_p\in\mathbb{C}$ are the roots of $f$ and that $a_1\notin\mathbb{R}$, $a_2\notin\mathbb{R}$, and $a_i\in\mathbb{R}$ for all $3\le i\le p$. Let $K=\mathbb{Q}(a_1,…,a_p)$ be the subfield of $\mathbb{C}$ generated by the roots of $f$. Consider the Galois group $G$ of $K$ over $\mathbb{Q}$ as a subgroup of $S_p$, the group of permutations of $\{a_1,…,a_p\}$. Which of the following statements are true?