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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)?

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