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    "title": "Wedderburn: every finite division ring is commutative",
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        "title": "A theorem on finite algebras",
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        "description": "Finiteness of the set and associativity are essential; the statement does not apply to infinite skew fields, finite nonassociative semifields, or arbitrary finite rings with zero divisors. This finite-division-ring theorem is not the separate 1907 semisimple-algebra structure theorem. The original itself credits a Dickson proof via canonical forms; no exclusive or unqualified priority claim is made. A generic-polynomial specialization in the first printed argument needs care over finite fields; a nonzero polynomial need not take a nonzero value there. The theorem is not a certification that every printed proof step is valid. The paper’s other arguments and Dickson attribution remain distinct.",
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          "citation": "A theorem on finite algebras",
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      "statement": "# Wedderburn: every finite division ring is commutative\n\nMultiplication need not commute in every algebra: quaternion multiplication is a familiar counterexample. Wedderburn’s 1905 theorem says that a finite associative division ring cannot have that behaviour. If each nonzero element has a multiplicative inverse and the set is finite, multiplication must commute, making the structure a finite field. The paper presents several arguments and acknowledges Dickson’s alternative proof; it is separate from the later structure theorem for semisimple algebras.\n\n## Scientific scope\n\nFiniteness of the set and associativity are essential; the statement does not apply to infinite skew fields, finite nonassociative semifields, or arbitrary finite rings with zero divisors. This finite-division-ring theorem is not the separate 1907 semisimple-algebra structure theorem. The original itself credits a Dickson proof via canonical forms; no exclusive or unqualified priority claim is made. A generic-polynomial specialization in the first printed argument needs care over finite fields; a nonzero polynomial need not take a nonzero value there. The theorem is not a certification that every printed proof step is valid. The paper’s other arguments and Dickson attribution remain distinct.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nJ. H. Maclagan-Wedderburn\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [A theorem on finite algebras](https://www.ams.org/journals/tran/1905-006-03/S0002-9947-1905-1500717-7/S0002-9947-1905-1500717-7.pdf)\n",
      "summary": "Multiplication need not commute in every algebra: quaternion multiplication is a familiar counterexample. Wedderburn’s 1905 theorem says that a finite associative division ring cannot have that behaviour. If each nonzero element has a multiplicative inverse and the set is finite, multiplication must commute, making the structure a finite field. The paper presents several arguments and acknowledges Dickson’s alternative proof; it is separate from the later structure theorem for semisimple algebras.",
      "title": "Wedderburn: every finite division ring is commutative"
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    "id": "wedderburn-finite-division-ring-commutativity-1905",
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    "title": "Wedderburn: every finite division ring is commutative",
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    "what_changed": "Multiplication need not commute in every algebra: quaternion multiplication is a familiar counterexample. Wedderburn’s 1905 theorem says that a finite associative division ring cannot have that behaviour. If each nonzero element has a multiplicative inverse and the set is finite, multiplication must commute, making the structure a finite field. The paper presents several arguments and acknowledges Dickson’s alternative proof; it is separate from the later structure theorem for semisimple algebras.",
    "scope": "Multiplication need not commute in every algebra: quaternion multiplication is a familiar counterexample. Wedderburn’s 1905 theorem says that a finite associative division ring cannot have that behaviour. If each nonzero element has a multiplicative inverse and the set is finite, multiplication must commute, making the structure a finite field. The paper presents several arguments and acknowledges Dickson’s alternative proof; it is separate from the later structure theorem for semisimple algebras. Finiteness of the set and associativity are essential; the statement does not apply to infinite skew fields, finite nonassociative semifields, or arbitrary finite rings with zero divisors. This finite-division-ring theorem is not the separate 1907 semisimple-algebra structure theorem. The original itself credits a Dickson proof via canonical forms; no exclusive or unqualified priority claim is made. A generic-polynomial specialization in the first printed argument needs care over finite fields; a nonzero polynomial need not take a nonzero value there. The theorem is not a certification that every printed proof step is valid. The paper’s other arguments and Dickson attribution remain distinct.",
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