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