Field of Characteristic Zero has Unique Prime Subfield

Theorem
Let $F$ be a field, whose zero is $0_F$ and whose unity is $1_F$, with characteristic zero.

Then there exists a unique $P \subseteq F$ such that:


 * 1) $P$ is a subfield of $F$;
 * 2) $P$ is isomorphic to the field of rational numbers $\left({\Q, +, \times}\right)$.

That is, $P \cong \Q$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$.

This field $P$ is called the prime subfield of $F$.

Proof
Follows directly from Subring Generated by Unity of Ring with Unity and Quotient Theorem for Monomorphisms.

Alternatively:


 * Let $\left({F, +, \circ}\right)$ be a field such that $\operatorname{Char} \left({F}\right) = 0$.

We can consistently define a mapping $\phi: \Q \to F$ by:

$\forall m, n \in \Z: n \ne 0: \phi \left({m / n}\right) = \left({m \cdot 1_F}\right) \circ \left({n \cdot 1_F}\right)^{-1}$.

By the ring theory analogue of Monomorphism Image Isomorphic to Domain, it follows that $P = \operatorname{Im} \left({\phi}\right)$ is a subfield of $F$ such that $P \cong \Q$.


 * Let $K$ be a subfield of $F$, and $P = \operatorname{Im} \left({\phi}\right)$ as defined above.

We know that $1_F \in K$.

Thus $K$ contains a subfield $P$ such that $P$ is isomorphic to $\Q$.


 * The uniqueness of $P$ follows from the fact that if $P_1$ and $P_2$ are both minimal subfields of $F$, then $P_1 \subseteq P_2$ and $P_2 \subseteq P_1$, thus $P_1 = P_2$.