Field has Prime Subfield
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Theorem
Let $\struct {F, +, \times}$ be a field.
Then $F$ has a subfield which is a prime field.
Proof
By definition of field, $F$ is a division ring where $\times$ is commutative.
Therefore all division subrings of $F$ are in fact subfields of $F$.
By Intersection of All Division Subrings is Prime Subfield, the intersection of all subfields of $F$ is a prime field which is a subfield of $F$.
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties