# Galois Field/Examples/Order 4

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## Example of Galois Field

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

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

## Proof

From Field with 4 Elements has only Order 2 Elements we have that a Galois field of order $4$, if it exists, must have this structure:

- $\struct {\GF, +}$ is the Klein $4$-group

- $\struct {\GF^*, \times}$ is the cyclic group of order $3$.

We have that $4 = 2^2$, and $2$ is prime.

From Galois Field of Order q Exists iff q is Prime Power, there exists at least one Galois field of order $4$.

As $\struct {\GF^*, \times}$ is the only such algebraic structure that can be a Galois field, it follows that $\struct {\GF^*, \times}$ *must* be a Galois field.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $25$