Primitive polynomial (field theory)

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

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