Subfield Test/Four-Step

Theorem

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

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

$\struct {K, +, \circ}$ is a subfield of $\struct {F, +, \times}$ if and only if these all hold:

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

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

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

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

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

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

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

$\Box$

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

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

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

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

$\blacksquare$

1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 56.3$ Subrings and subfields