Subfield Test

Theorem
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$.

Let $K$ be a subset of $F$.

Then $\struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$ these all hold:


 * $(1): \quad K^* \ne \O$


 * $(2): \quad \forall x, y \in K: x + \paren {-y} \in K$


 * $(3): \quad \forall x, y \in K: x \circ y \in K$


 * $(4): \quad x \in K^* \implies x^{-1} \in K^*$

where $K^*$ denotes $K \setminus \set {0_F}$.

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

Then the conditions $(1)$ to $(4)$ all hold by virtue of the field axioms.

Sufficient Condition
Suppose the conditions $(1)$ to $(4)$ hold.

From the Division Subring Test, it follows that $\struct {K, +, \circ}$ is a division ring.

As $\struct {F, +, \circ}$ is a field, then $\circ$ is commutative on all of $F$.

Therefore $\circ$ is commutative also on $K$ by Restriction of Commutative Operation is Commutative.

Thus $\struct {K, +, \circ}$ is a field.