Celebratio Mathematica — Dickson

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Works connected to John Charles Fields

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L. E. Dickson: “Outline of the theory to date of the arithmetics of algebras,” pp. 95–102 in Proceedings of the International Mathematical Congress, 1924 (Toronto, 11–16 August, 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press, 1928. JFM 54.0160.03 incollection

L. E. Dickson: “Further development of the theory of arithmetics of algebras,” pp. 173–184 in Proceedings of the International Mathematical Congress, 1924 (Toronto, 11–16 August, 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press, 1928. JFM 54.0161.01 incollection

L. E. Dickson: “A new theory of linear transformations and pairs of bilinear forms,” pp. 361–363 in Proceedings of the International Mathematical Congress, 1924 (Toronto, 11–16 August, 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press (Toronto), 1928. JFM 54.0110.06 incollection

It is customary to develop the theory of pairs of bilinear forms having the matrices \( M \) and \( N \), and by considering the special case in which \( N \) is the identity (or unit) matrix \( I \) to deduce the theory of the canonical form of a linear transformation \( T \). We here proceed in reverse order and first develop independently a simple theory of linear transformation and later deduce the theory of equivalence of pairs of matrices and hence of pairs of bilinear forms. We avoid the introduction of irrationalities and employ only rational processes, so that our theory holds for any given field (or domain of rationality). We obtain a simple interpretation of invariant factors.

Moreover, we avoid the consideration of matrices whose elements are any polynomials in a variable \( \lambda \) as well as elementary transformations of matrices.