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Published online by Cambridge University Press: 18 May 2009
The purpose of this paper is to present a proof of the following theorem of Maclagan-Wedderburn.*
Every finite skew-field† is a field.
The proof depends on group theory and on the properties of Galois fields. As an introduction, §§1–4 are devoted to a systematic and self-contained account of the theory of Galois fields.
* A Theorem on Finite Algebras. American M. S. Transactions, 6, pp. 349–352, (1905).Google Scholar
† A skew-field or division ring is an algebraic system which satisfies all the postulates of a field except possibly that which demands that multiplication shall be commutative; i.e., it is a ring, not necessarily commutative, whose non-zero elements form a multiplicative group. The theorem states that if the number of elements is finite, the commutative property of multiplication is a consequence of the other postulates.
‡ Liouville's Journal XI (1846), pp. 381–444.Google Scholar
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