Galois Field/Examples

Examples of Galois Fields

Order $4$ Galois Field

The algebraic structure $\struct {\GF, +, \times}$ defined by the following Cayley tables is a Galois field:

$\begin{array} {c|cccc}

+ & 0 & 1 & a & b \\ \hline 0 & 0 & 1 & a & b \\ 1 & 1 & 0 & b & a \\ a & a & b & 0 & 1 \\ b & b & a & 1 & 0 \\ \end{array} \qquad \begin{array} {c|cccc} \times & 0 & 1 & a & b \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & a & b \\ a & 0 & a & b & 1 \\ b & 0 & b & 1 & a \\ \end{array}$

Field of Integers Modulo Prime

The field of integers modulo $p$ is a Galois field:

Let $p \in \Bbb P$ be a prime number.

Let $\Z_p$ be the set of integers modulo $p$.

Let $+_p$ and $\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively.

The algebraic structure $\struct {\Z_p, +_p, \times_p}$ is the field of integers modulo $p$.