Definition:Galois Field

Definition

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

The symbol conventionally used to denote a Galois field of $p$ elements is $\map \GF p$.

Also known as

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

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

Examples

Order $4$ Galois Field

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

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

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 \(\GF\) is \GF  or \Bbb F or \mathbb F.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: Galois field (finite field)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Galois field (finite field)