Definition:Galois Field

Definition

A Galois field $\struct {F, +, \circ}$ is a field such that $F$ is a finite set.

The symbol conventionally used to denote a Galois field is $\F$.

Also known as

Some sources do not mention Galois, but merely refer to a finite field.

Some sources use the notation $\map {\operatorname {GF} } n$ to denote a Galois field of order $n$.

Examples

Order $4$ Galois Field

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

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

Also see

• Definition:Infinite Field

• Results about Galois fields can be found here.

Source of Name

This entry was named for Évariste Galois.

Technical Note

The $\LaTeX$ code for \(\F\) is \F  or \FF.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): $\S 1.1$
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts