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 $\struct {\set {0_R, 1_R}, +, \circ}$:


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

\struct {\set {0_R, 1_R}, +} & 0_R & 1_R \\ \hline 0_R & 0_R & 1_R \\ 1_R & 1_R & 0_R \\ \end{array} \qquad \begin{array} {r|rr} \struct {\set {0_R, 1_R}, \circ} & 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 $2$ $\struct {\Z_2, +_2, \times_2}$:

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