Field has Prime Subfield

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