Smallest Field is Field

Theorem
The ring $\left({\left\{{0_R, 1_R}\right\}, +, \circ}\right)$ 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 $\left({\left\{{0_R, 1_R}\right\}, +, \circ}\right)$ to be a field:


 * $\left({\left\{{0_R, 1_R}\right\}, +}\right)$ must be an abelian group. This is fulfilled as this is the parity group.


 * $\left({\left\{{0_R, 1_R}\right\}, \circ}\right)$ must be a commutative division ring. This is fulfilled, as $\left({\left\{{0_R, 1_R}\right\}^*, \circ}\right) = \left({\left\{{1_R}\right\}, \circ}\right)$ 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.