# Definition:Subfield/Proper Subfield

## Definition

Let $\struct {K, +, \circ}$ be a subfield of $\struct {F, +, \circ}$.

Then $\struct {K, +, \circ}$ is a proper subfield of $\struct {F, +, \circ}$ if and only if $K \ne F$.

That is, $\struct {K, +, \circ}$ is a proper subfield of $\struct {F, +, \circ}$ if and only if:

$(1): \quad \struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$
$(2): \quad K$ is a proper subset of $F$.