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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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It's a guess only:

what about taking $\mathfrak M$ number of transcendentals $(\xi_i)_{i<\mathfrak M}$ over $\mathbb Q$, and consider the algebraic closure of $\mathbb Q(\xi_i)_i$?

Edit: Instead of the algebraic closure, consider only (all the roots of) all irreducible polynomials that has at least one transcendental over $\mathbb Q$ among its coefficients..

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    You're right. As I indicated, it was a guess only.2012-09-24