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  "body": "\n## Summary\n\nThe Laurent polynomial ring $R[x, x^{-1}]$ over a [[ring|0B1261B5ED]] $R$ consists of finite sums $\\sum a_i x^i$ in which the exponents $i$ may be negative.\nIt is the [[monoid ring|1HM7ZYMPD7]] of the additive group $\\Z$ over $R$.\nIn it the variable $x$ is a [[unit|0XVEZDE9J8]] with inverse $x^{-1}$.\nIt is the group ring of the infinite cyclic group.\nOver a field it is the coordinate ring of the line with the origin removed.\n\n## Explanations\n\n### L1 human\n\nA Laurent polynomial is a polynomial that may also contain negative powers of the variable, like $3x^{-2} + 1 + 5x$.\nThere are still only finitely many terms.\nAdding and multiplying works as for polynomials, with the rule $x^i x^j = x^{i+j}$ for all integers $i, j$.\nFor example, $(x + x^{-1})^2 = x^2 + 2 + x^{-2}$.\nThe point of the construction is that $x$ now has an inverse, which it does not have among ordinary polynomials.\n",
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