# Intersection of Subfields is Largest Subfield Contained in all Subfields

## Theorem

Let $\struct {F, +, \circ}$ be a field.

Let $\mathbb K$ be a non-empty set of subfields of $F$.

Let $\bigcap \mathbb K$ be the intersection of the elements of $\mathbb K$.

Then $\bigcap \mathbb K$ is the largest subfield of $F$ contained in each element of $\mathbb K$.

## Proof

Let $L = \bigcap \mathbb K$.

From Intersection of Subfields is Subfield, $\struct {L, +, \circ}$ is itself a subfield of $\struct {F, +, \circ}$.

A field is by definition also a division subring.

Thus $\struct {L, +, \circ}$ is the largest division subring of $F$ contained in each element of $\mathbb K$.

But as $\struct {F, +, \circ}$ is a field, $\circ$ is commutative.

So by Restriction of Commutative Operation is Commutative, $\struct {L, +, \circ}$ is also a field.

Hence the result.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$: Theorem $23.1$