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 $$P_1 = P_2$$.