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  "body": "\n## Summary\n\nA primitive polynomial over a [[finite field|0ZXZ82FTX2]] $\\F_q$ is the minimal polynomial of a [[primitive element|00QSK85Y2X]] of some extension $\\F_{q^m}$.\nWith such a polynomial, every nonzero element of $\\F_{q^m}$ is a power of one root, which makes multiplication easy to compute.\nPrimitive polynomials over $\\F_2$ give linear feedback shift registers of maximal period.\nThis notion differs from the [[primitive polynomial|0TXNRQZY0H]] of ring theory, whose coefficients have no common factor.\n\n## Explanations\n\n### L1 human\n\nTake a root $\\alpha$ of an irreducible polynomial and look at its powers $\\alpha, \\alpha^2, \\alpha^3, \\dots$.\nThe polynomial is primitive when these powers run through every nonzero element of the field before they come back to $1$.\nOver $\\F_2$, the polynomial $x^2 + x + 1$ is primitive: its root $\\omega$ gives $\\omega$, $\\omega^2 = \\omega + 1$ and $\\omega^3 = 1$, which are the three nonzero elements of $\\F_4$.\n",
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