Let $\mathfrak{M}$ be an infinite cardinal. Consider all fields $F$ which have the following properties:
(1) $F$ contains $\mathbb{Q}$.
(2) $F$ has cardinality $\leqslant \mathfrak{M}$.
(3) All elements of $F \setminus \mathbb{Q}$ are transcendental over $\mathbb{Q}$.
(Such a field need not be a purely transcendental extension of $\mathbb{Q}$.)
Does there exist a field that satisfies (1)-(3) and contains an isomorphic copy of any field which has properties (1)-(3)?