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  "body": "\n## Summary\n\nFinite field arithmetic is the way to add, multiply and divide elements of a [[finite field|0ZXZ82FTX2]] in practice.\nThe usual model of $\\F_{p^n}$ is the set of polynomials of degree less than $n$ over $\\F_p$.\nOne computes with them modulo a fixed [[irreducible polynomial|03FKX7Z5DC]] of degree $n$.\nIt is the basis of error-correcting codes and of much of modern cryptography.\n\n## Explanations\n\n### L1 human\n\nIn $\\F_p$ one computes with integers modulo a prime $p$.\nFor a field with $p^n$ elements, one computes with polynomials instead, and one reduces modulo a fixed polynomial, as one reduces integers modulo $p$.\nOver $\\F_2$, an element of $\\F_{2^8}$ is a string of $8$ bits, and addition is the bitwise exclusive or.\nFor example, in $\\F_8$ built with $x^3 + x + 1$, one has $x^3 = x + 1$, so $(x^2 + 1)(x + 1) = x^3 + x^2 + x + 1 = x^2$.\n",
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