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A Group-Theoretic Proof of a Theorem of MacLagan-Wedderburn

Published online by Cambridge University Press:  18 May 2009

Hans J. Zassenhaus
Hans J. Zassenhaus
Affiliation:
McGill University, Montreal, Canada.
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Extract

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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 1 , Issue 2 , July 1952 , pp. 53 - 63
Copyright
Copyright © Glasgow Mathematical Journal Trust 1952

References

* 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