Symmetric and alternate matrices in an arbitrary field. I
Alexa A. Albert · Transactions of the American Mathematical Society · 1938
p. 894 for the definition and some elementary properties of involutions.t The proofs of these lemmas may be found in the author's Modern Higher Algebra, chap.5, University of Chicago Press, 1937.* We shall henceforth use the notation diag[Gi, • • • , Cs] for (2) to simplify printing, t See the author's Modern Higher Algebra, chap. 5. ■ * Cf. the author's Modern Higher Algebra.The theory is almost exactly the same as in L. E. Dickson's Modern Algebraic Theories.f A perfect field 5 of characteristic two has the property that every a of % is equal to b2, b in %.Such fields with an over-field Ä = 3(0) exist.For the definition see van der Waeiden'sModerne Algebra, vol. 1, as well as the author's own Modern Higher Algebra.* This is the standard technique for the study of congruences f(x) = 0 (mod p) in the theory of numbers (as in L. E. Dickson's Introduction to the Theory of Numbers, p. 16, ex.4).We are using the analogous property of the polynomial domain (cf.Lemma 35.21, p. 60, of MacDuffee's tract on The Theory of Matrices).* See the author's paper, Involutorial simple algebras and real Riemann matrices, loc.cit.t Note that conversely / determines E only up to a scalar factor.* N. Jacobson, A class of normal simple Lie algebras of characteristic zero, Annals of Mathematics, vol.38 (1937), pp.508-517.Note that conversely the property in the preceding footnote implies that Ea defines an involution cogredient with that defined by E if and only if E¡, is congruent to a scalar multiple of E.