Field of Characteristic Zero has Unique Prime Subfield/Proof 1

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:

$(1): \quad P$ is a subfield of $F$

$(2): \quad P$ is isomorphic to the field of rational numbers $\struct {\Q, +, \times}$.

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

Quotient Theorem for Monomorphisms

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.10$