Editing Primitive polynomial (field theory) (section)

==Examples==
Over {{math|GF(3)}} the polynomial {{math|''x''<sup>2</sup> + 1}} is irreducible but not primitive because it divides {{math|''x''<sup>4</sup> − 1}}: its roots generate a cyclic group of order 4, while the multiplicative group of {{math|GF(3<sup>2</sup>)}} is a cyclic group of order 8. The polynomial {{math|''x''<sup>2</sup> + 2''x'' + 2}}, on the other hand, is primitive. Denote one of its roots by {{mvar|α}}. Then, because the natural numbers less than and relatively prime to {{math|3<sup>2</sup> − 1 {{=}} 8}} are 1, 3, 5, and 7, the four primitive roots in {{math|GF(3<sup>2</sup>)}} are {{mvar|α}}, {{math|''α''<sup>3</sup> {{=}} 2''α'' + 1}}, {{math|''α''<sup>5</sup> {{=}} 2''α''}}, and {{math|''α''<sup>7</sup> {{=}} ''α'' + 2}}. The primitive roots {{mvar|α}} and {{math|''α''<sup>3</sup>}} are algebraically conjugate. Indeed {{math|''x''<sup>2</sup> + 2''x'' + 2 {{=}} (''x'' − ''α'') (''x'' − (2''α'' + 1))}}. The remaining primitive roots {{math|''α''<sup>5</sup>}} and {{math|''α''<sup>7</sup> {{=}} (''α''<sup>5</sup>)<sup>3</sup>}} are also algebraically conjugate and produce the second primitive polynomial: {{math|''x''<sup>2</sup> + ''x'' + 2 {{=}} (''x'' − 2''α'') (''x'' − (''α'' + 2))}}.

For degree 3, {{math|GF(3<sup>3</sup>)}} has {{math|''φ''(3<sup>3</sup> − 1) {{=}} ''φ''(26) {{=}} 12}} primitive elements. As each primitive polynomial of degree 3 has three roots, all necessarily primitive, there are {{math|12 / 3 {{=}} 4}} primitive polynomials of degree 3. One primitive polynomial is {{math|''x''<sup>3</sup> + 2''x'' + 1}}. Denoting one of its roots by {{mvar|γ}}, the algebraically conjugate elements are {{math|''γ''<sup>3</sup>}} and {{math|''γ''<sup>9</sup>}}. The other primitive polynomials are associated with algebraically conjugate sets built on other primitive elements {{math|''γ''<sup>''r''</sup>}} with {{mvar|r}} relatively prime to 26:
:<math>\begin{align}x^3+2x+1 & = (x-\gamma)(x-\gamma^3)(x-\gamma^9)\\
x^3+2x^2+x+1 &= (x-\gamma^5)(x-\gamma^{5\cdot3})(x-\gamma^{5\cdot9}) = (x-\gamma^5)(x-\gamma^{15})(x-\gamma^{19})\\
x^3+x^2+2x+1 &= (x-\gamma^7)(x-\gamma^{7\cdot3})(x-\gamma^{7\cdot9}) = (x-\gamma^7)(x-\gamma^{21})(x-\gamma^{11})\\
x^3+2x^2+1 &= (x-\gamma^{17})(x-\gamma^{17\cdot3})(x-\gamma^{17\cdot9}) = (x-\gamma^{17})(x-\gamma^{25})(x-\gamma^{23}).
\end{align}</math>