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  "body": "\n## Summary\n\nA Conway polynomial $C_{p,n}$ is a fixed choice of [[primitive polynomial|HQ7KMZYPSQ]] of degree $n$ over $\\F_p$.\nThe choices for different $n$ are compatible, so the fields $\\F_{p^n}$ built from them fit together as subfields.\nComputer algebra systems use Conway polynomials as a standard model of [[finite fields|0ZXZ82FTX2]], so that results from different programs agree.\n\n## Explanations\n\n### L1 human\n\nThere are many ways to build the field with $p^n$ elements, one for each irreducible polynomial of degree $n$.\nA Conway polynomial picks one of them by a fixed rule: the first in a dictionary order among the primitive polynomials that fit with the smaller choices.\nFor $p = 2$, the first ones are $x + 1$, $x^2 + x + 1$, $x^3 + x + 1$ and $x^4 + x + 1$.\n",
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