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 $K \setminus \set {0_F}$.
Proof
Necessary Condition
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.
$\Box$
Sufficient Condition
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 $F$.
From Restriction of Commutative Operation is Commutative, $\times$ is commutative also on $K$.
Thus $\struct {K, +, \times}$ is a field.
$\blacksquare$
Sources
- 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