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.