Definition:Smallest Field
Definition
The smallest field is the set of integers modulo $2$ under modulo addition and modulo multiplication:
- $\struct {\Z_2, +_2, \times_2}$
This field has $2$ elements.
Cayley Tables
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 $2$ $\struct {\Z_2, +_2, \times_2}$:
- $\begin{array} {r|rr}
\struct {\Z_2, +_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array} \qquad \begin{array}{r|rr} \struct {\Z_2, \times_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 0 2 \\ \eqclass 1 2 & \eqclass 0 2 & \eqclass 1 2 \\ \end{array}$
They can be presented more simply as:
- $\begin{array}{r|rr}
+ & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{r|rr} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$
Parity Ring
It can also be expressed in terms of integer parity:
- $\begin{array}{r|rr}
+ & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array} \qquad \begin{array}{r|rr} \times & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{even} \\ \text{odd} & \text{even} & \text{odd} \\ \end{array}$
Also see
- Smallest Field is Field, where it is proved that $\struct {\Z_2, +_2, \times_2}$ is a field.