# Field of Characteristic Zero has Unique Prime Subfield/Proof 1

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## 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:

$\blacksquare$

## Sources

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