Set of Subfields forms Complete Lattice

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

Let $\mathbb F$ be the set of all subfields of $F$.

Then $\struct {\mathbb F, \subseteq}$ is a complete lattice.

Proof
Let $\O \subset \mathbb S \subseteq \mathbb F$.

By Intersection of Subfields is Largest Subfield Contained in all Subfields:
 * $\bigcap \mathbb S$ is the largest subfield of $F$ contained in each of the elements of $\mathbb S$.

By Intersection of Subfields Containing Subset is Smallest:
 * The intersection of the set of all subfields of $F$ containing $\bigcup \mathbb S$ is the smallest subfield of $F$ containing $\bigcup \mathbb S$.

Thus:
 * Not only is $\bigcap \mathbb S$ a lower bound of $\mathbb S$, but also the largest, and therefore an infimum.


 * The supremum of $\mathbb S$ is the intersection of the set of all subfields of $R$.

Therefore $\struct {\mathbb F, \subseteq}$ is a complete lattice.