Subfield Test

Theorem
Let $\left({F, +, \circ}\right)$ be a field, and let $K$ be a subset of $F$.

Then $\left({K, +, \circ}\right)$ is a subfield of $\left({F, +, \circ}\right)$ iff these all hold:


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


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


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


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

Necessary Condition
Let $\left({K, +, \circ}\right)$ be a subfield of $\left({F, +, \circ}\right)$.

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 $\left({K, +, \circ}\right)$ is a division ring.

As $\left({F, +, \circ}\right)$ is a field, then $\circ$ is commutative on all of $F$, and therefore also on $K$ by Restriction of Operation Commutativity.

Thus $\left({K, +, \circ}\right)$ is a field.