Editing Finite field (section)

=== GF(8) and GF(27) ===
The polynomial
<math display="block">X^3-X-1</math>
is irreducible over <math>\mathrm{GF}(2)</math> and <math>\mathrm{GF}(3)</math>, that is, it is irreducible [[modulo]] <math>2</math> and <math>3</math> (to show this, it suffices to show that it has no root in <math>\mathrm{GF}(2)</math> nor in <math>\mathrm{GF}(3)</math>). It follows that the elements of <math>\mathrm{GF}(8)</math> and <math>\mathrm{GF}(27)</math> may be represented by [[expression (mathematics)|expressions]]
<math display="block">a+b\alpha+c\alpha^2,</math>
where <math>a, b, c</math> are elements of <math>\mathrm{GF}(2)</math> or <math>\mathrm{GF}(3)</math> (respectively), and <math>\alpha</math> is a symbol such that 
<math display="block">\alpha^3=\alpha+1.</math>

The addition, additive inverse and multiplication on <math>\mathrm{GF}(8)</math> and <math>\mathrm{GF}(27)</math> may thus be defined as follows; in following formulas, the operations between elements of <math>\mathrm{GF}(2)</math> or <math>\mathrm{GF}(3)</math>, represented by Latin letters, are the operations in <math>\mathrm{GF}(2)</math> or <math>\mathrm{GF}(3)</math>, respectively:
<math display="block">
\begin{align}
-(a+b\alpha+c\alpha^2)&=-a+(-b)\alpha+(-c)\alpha^2 \qquad\text{(for } \mathrm{GF}(8), \text{this operation is the identity)}\\
(a+b\alpha+c\alpha^2)+(d+e\alpha+f\alpha^2)&=(a+d)+(b+e)\alpha+(c+f)\alpha^2\\
(a+b\alpha+c\alpha^2)(d+e\alpha+f\alpha^2)&=(ad + bf+ce)+ (ae+bd+bf+ce+cf)\alpha+(af+be+cd+cf)\alpha^2
\end{align}
</math>