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 $q$ elements is $\map \GF q$.


Also denoted as

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

Some sources use $\GF_q$ for $\map \GF q$.

Some sources place further emphasis on the formal nature of a Galois field by denoting it $\map {\mathrm {GF} } {p^n}$ or $\map \GF {p^n}$, and so on.


Also known as

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


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}$


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 $p$.


Also see

  • Results about Galois fields can be found here.


Source of Name

This entry was named for Évariste Galois.


Historical Note

The study of Galois fields was initiated by Évariste Galois in $1830$.


Technical Note

The $\LaTeX$ code for \(\GF\) is \GF  or \Bbb F or \mathbb F.


Sources

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