A theorem on finite algebras
J. H. Maclagan-Wedderburn · Transactions of the American Mathematical Society · 1905
FrobeniusI and C. S. Peirce § have shown that, in the domain of all real numbers, the only linear associative algebras every number of which, except zero, possesses an inverse, are quaternions and its subalgebras, and also that in the complex domain no algebra has that property.In the present paper it is shown that the Galois field is the only algebra of this type which possesses a finite number of elements.II.