Definition:Galois Field
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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 of $p$ elements is $\map \F p$.
Also known as
Some sources do not mention Galois, but merely refer to a finite field.
Some sources use the notation $\map {\operatorname {GF} } p$ 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
- 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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Galois field (finite field)
