Smallest Field is Field

Theorem
The ring $\struct {\set {0_R, 1_R}, +, \circ}$ is the smallest algebraic structure which is a field.

Proof
The null ring, which contains one element, is not a field as it is trivial.

Therefore any field must contain at least two elements.

For $\struct {\set {0_R, 1_R}, +, \circ}$ to be a field:


 * $\struct {\set {0_R, 1_R}, +}$ must be an abelian group.

This is fulfilled as this is the parity group.


 * $\struct {\set {0_R, 1_R}, \circ}$ must be a commutative division ring.

This is fulfilled, as $\struct {\set {0_R, 1_R}^*, \circ} = \struct {\set {1_R}, \circ}$ is the trivial group.


 * $\circ$ needs to distribute over $+$. This follows directly from Ring Product with Zero and the behaviour of the identity element in a group.