Definition:Galois Field

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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 $n$.

Some sources use $\GF_p$ for $\map \GF p$.


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

  • 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

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