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  "body": "\n## Summary\n\nA Hahn series is a formal sum $\\sum_\\gamma a_\\gamma t^\\gamma$ with coefficients in a field and exponents in a [[linearly ordered group|P2E8BNYC7B]].\nIts set of exponents must be [[well-ordered|01CGSJ0GNQ]].\nHahn series generalise [[formal power series|033WGN00BF]] and [[formal Laurent series|YZWYF7DXTW]] to exponents that may be fractions, reals, or elements of any ordered abelian group.\nThey form a field with a natural [[valuation|B85T5VKR3M]], and these fields are a main source of examples of valued and ordered fields.\n\n## Explanations\n\n### L1 human\n\nA Laurent series in $t$ uses integer powers, starting from some lowest power and going up.\nA Hahn series allows any exponents from an ordered group, like $t^{1/2}$ or $t^{\\sqrt{2}}$, as long as each nonempty set of exponents used has a smallest element.\nFor example $t^{1/2} + t^{2/3} + t^{3/4} + \\cdots$ is a Hahn series with rational exponents.\nThe rule about smallest elements is exactly what makes products of two such series well defined.\n",
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