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  "body": "\n## Summary\n\nA Galois ring is a finite [[commutative ring|0310REJJY8]] that extends $\\Z/p^n\\Z$ in the same way as a [[finite field|0ZXZ82FTX2]] $\\F_{p^r}$ extends $\\F_p$.\nGalois rings are [[local rings|MZ02BKPHJA]], and their residue fields are finite fields.\nThey contain both the rings $\\Z/p^n\\Z$ and the finite fields as special cases.\nThey are used in the theory of codes over $\\Z/4\\Z$ and in the study of [[finite rings|X6713ZP2NH]].\n\n## Explanations\n\n### L1 human\n\nA finite field $\\F_{p^r}$ is built from arithmetic modulo a prime $p$ by adding a root of an irreducible polynomial.\nA Galois ring does the same over arithmetic modulo a prime power $p^n$.\nFor example, take arithmetic modulo $4$ and add a symbol $x$ with $x^2 = -x - 1$.\nThe result has $16$ elements $a + bx$ with $a, b \\in \\set{0, 1, 2, 3}$.\nReducing everything modulo $2$ gives the field $\\F_4$ with four elements.\n",
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