Primitive polynomial (field theory)

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In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p^m). This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p^m) such that ${\displaystyle \{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\ldots \alpha ^{p^{m}-2}\}}$ is the entire field GF(p^m). This implies that α is a primitive (p^m − 1)-root of unity in GF(p^m).