Editing Primitive polynomial (field theory) (section)

==Properties==
* Because all minimal polynomials are [[irreducible polynomial|irreducible]], all primitive polynomials are also irreducible.
* A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by&nbsp;''x''. Over [[GF(2)]], {{nowrap|''x'' + 1}} is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by {{nowrap|''x'' + 1}} (it has 1 as a root).
* An [[irreducible polynomial]] ''F''(''x'') of degree ''m'' over GF(''p''), where ''p'' is prime, is a primitive polynomial if the smallest positive integer ''n'' such that ''F''(''x'') divides {{nowrap|''x''<sup>''n''</sup> − 1}} is {{nowrap|1=''n'' = ''p''<sup>''m''</sup> − 1}}.
* A primitive polynomial of degree {{mvar|m}} has {{mvar|m}} different roots in {{math|GF(''p''<sup>''m''</sup>)}}, which all have [[order (group theory)|order]] {{math|''p''<sup>''m''</sup> − 1}}, meaning that any of them generates the [[Finite field#Multiplicative structure|multiplicative group]] of the field.
* Over GF(''p'') there are exactly {{math|''φ''(''p''<sup>''m''</sup> − 1)}} primitive elements and {{math|''φ''(''p''<sup>''m''</sup> − 1) / ''m''}} primitive polynomials, each of degree {{mvar|m}}, where {{mvar|φ}} is [[Euler's totient function]].<ref>Enumerations of primitive polynomials by degree over {{math|GF(2)}}, {{math|GF(3)}}, {{math|GF(5)}}, {{math|GF(7)}}, and {{math|GF(11)}} are given by sequences {{OEIS link|A011260}}, {{OEIS link|A027385}}, {{OEIS link|A027741}}, {{OEIS link|A027743}}, and {{OEIS link|A319166}} in the [[Online Encyclopedia of Integer Sequences]].</ref>
* The [[Conjugate element (field theory)|algebraic conjugates]] of a primitive element {{mvar|α}} in {{math|GF(''p''<sup>''m''</sup>)}} are {{mvar|α}}, {{math|''α''{{i sup|''p''}}}}, {{math|''α''{{i sup|''p''{{sup|2}}}}}}, …, {{math|''α''{{i sup|''p''{{sup|''m''−1}}}}}} and so the primitive polynomial {{math|''F''(''x'')}} has explicit form {{math|''F''(''x'') {{=}} (''x'' − ''α'') (''x'' − ''α''{{i sup|''p''}}) (''x'' − ''α''{{i sup|''p''{{sup|2}}}}) … (''x'' − ''α''{{i sup|''p''{{sup|''m''−1}}}})}}. That the coefficients of a polynomial of this form, for any {{mvar|α}} in {{math|GF(''p''<sup>''n''</sup>)}}, not necessarily primitive, lie in {{math|GF(''p'')}} follows from the property that the polynomial is invariant under application of the [[Frobenius automorphism]] to its coefficients (using {{math|''α''<sup>''p''<sup>''n''</sup></sup> {{=}} ''α''}}) and from the fact that the fixed field of the Frobenius automorphism is {{math|GF(''p'')}}.