Subfield Test

Theorem

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

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

Three-Step Subfield Test

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

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

$(2): \quad \forall x, y \in K: x - y \in K$

$(3): \quad \forall x \in K: \forall y \in K^*: \dfrac x y \in K$

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

Four-Step Subfield Test

$\struct {K, +, \circ}$ 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 $K \setminus \set {0_F}$.