Smallest Field/Cayley Tables

Cayley Tables for the Smallest Field
The smallest field can be completely described by showing its Cayley tables.

In purely abstract form as $\left({\left\{{0_R, 1_R}\right\}, +, \circ}\right)$:


 * $\begin{array}{r|rr}

\left({\left\{{0_R, 1_R}\right\}, +}\right) & 0_R & 1_R \\ \hline 0_R & 0_R & 1_R \\ 1_R & 1_R & 0_R \\ \end{array} \qquad \begin{array}{r|rr} \left({\left\{{0_R, 1_R}\right\}, \circ}\right) & 0_R & 1_R \\ \hline 0_R & 0_R & 0_R \\ 1_R & 0_R & 1_R \\ \end{array}$

Ring of Integers Modulo $2$
It can also be expressed as the ring of integers modulo $2$ $\left({\Z_2, +_2, \times_2}\right)$:

Parity Ring
It can also be expressed in terms of integer parity: