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  "body": "\n## Summary\n\nA semifield is an algebraic structure that satisfies all the axioms of a [[division ring|0EA5BTA0CX]] except possibly the associativity of multiplication.\nSemifields arise in finite geometry, where each semifield coordinatizes a projective plane, called a semifield plane.\nFinite semifields that are not fields give non-Desarguesian planes.\nIn other areas, a semifield is a semiring whose nonzero elements form a group under multiplication.\nThis node is about the first meaning.\n\n## Explanations\n\n### L1 human\n\nIn a field one can add, subtract, multiply and divide by any nonzero element, and multiplication is associative and commutative.\nA semifield keeps addition, subtraction and division, and the two distributive laws, but drops the rule $(ab)c = a(bc)$.\nEvery field and every division ring is a semifield.\nThe octonions, an eight-dimensional algebra over the real numbers, are a semifield whose multiplication is not associative.\n",
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