Smallest Field/Cayley Tables

Cayley Table for the Smallest Field
We can completely describe the smallest field by showing its Cayley tables.


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

\left({\Z_2, +_2}\right) & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2\\ \hline \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 \\ \end{array} \qquad \begin{array}{r|rr} \left({\Z_2, \times_2}\right) & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2\\ \hline \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 \\ \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \end{array}$

It can be expressed in terms of integer parity:
 * $\begin{array}{r|rr}

+ & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array} \qquad \begin{array}{r|rr} \times & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{even} \\ \text{odd} & \text{even} & \text{odd} \\ \end{array}$

It can also be expressed completely abstractly 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}$