I have already shown existence, but I have no idea how to do this because I am not allowed to use " For any collection of $P \subseteq F[x]$ of non-zero polynomials over a field $F$, a splitting field of $P$ over $F$ exists and is unique up to isomorphism of field extensions; that is, if $G_{1}$ and $G_{2}$ are two splitting fields of $P$ over $F$ then there exists a field isomorphism $\phi: G_{1} \to G_{2}$ with the following commutative diagram where $F \to G_{1}$ and $F \to G_{2}$ are the canonical embeddings:
I am assuming we take another field to be the algebraic closure of $\mathbb{R}$ and arrive at some contradiction, but I'm at a loss