訂正日: 2006/08/29訂正理由: -訂正箇所: 引用文献情報訂正内容: Wrong : 1) W. K. Clifford, Amer. J. Math., 1 (1878). A. S. Eddington, J. London Math. Soc., 7 (1932), 8(1933). M. H. A. Newmann, J. London Math. Soc. 7(1932). J. E. Littlewood, J. London Math. Soc. 9(1934). R. Brauer, H. Weyl, Amer. J. Math., 57(1935), H. Weyl, Classical groups, (1939). D. Wajnsztejn, Studia. Math., 9(1940). C. Chevalley, Theory of Lie groups, (1946). H. C. Lee, Ann. of Math., 49(1948). For the representations by real orthogonal matrices cf. A. Hurwitz (Werke II), J. Radon, Abh. Math. Sem. Hamburg, 1(1923), B. Eckmann, Comment. Math. Helv., 15(1943). 2). Cf. A. S. Eddington and M. H. A. Newmann, loc. cit. 1). Theorem 2 of Newmann's paper is incomplete, cf. Theorem 4 of this note. 3) Cf. C. Chevalley, loc. cit. 1). 4) In fact for a prime field K with χ(K)=p≠2 there are α, β such that 1+α2+β2=0. Let p=2m+1 and let α1, ... αm, be a system of representatives of quadratic rests mod. p. If α2+ β2+1 were_??_0(mod. p) for any, α β then 0, α1, ..., αm, -1-αm (mod. p) would be a complete system of representatatives mod p. If we add them all, then -m≡1+2+...+(p-1) ≡0 (mod. p), which is a contradiction. 5) In fact 1+λ2≡0 (mod. p) has a solution if and only if p≡1 (mod. 4). 6) Cf. L. E. Dickson, Algebren und ihre Zahlentheorie (1927). or R. Brauer-E. Noether, S. B. Preuss. Akad., (1927). See also Appendix I. Cf also A. A. Albert, Structure of algebras, (1939), p. 147, Theorem 27. 7) This follows immediately from a well-known theorem of algebra (for example M. Deuring, Algebren (1934), p. 58, Satz 1). See also Appendix II. 8) See Appendix, III. Cf. also A.A. Albert, loc. cit 6). 9) See Appendix IV. 10) See Appendix V. 11) For the case when K is the complex number field this result is given by M. H. A. Newmann, loc. cit. 1). 12) For the case when Λ is the real number field, this result is given by A. S. Eddington for m=2, and by M. H. A. Newmann for m=3. For m>3 the result of Newmann (Theorem 2) is incomplete. He misses the possibility of the case where all matrices are purely imaginary.
Right : 1) W. K. Clifford, Amer. J. Math., 1 (1878). A. S. Eddington, J. London Math. Soc., 7 (1932), 8 (1933). M. H. A. Newmann, Ibid. 7 (1932). J. E. Littlewood, Ibid. 9 (1934). R. Brauer, H. Weyl, Amer. J. Math., 57 (1935), H. Weyl, Classical groups, (1939). D. Wajnsztejn, Studia. Math., 9 (1940). C. Chevalley, Theory of Lie groups, (1946). H. C. Lee, Ann. of Math., 49 (1948). For the representations by real orthogonal matrices cf. A. Hurwitz (Werke II), J. Radon, Abh. Math. Sem. Hamburg, 1(1923), B. Eckmann, Comment. Math. Helv., 15(1943). 2). Cf. A. S. Eddington and M. H. A. Newmann, loc. cit. 1). Theorem 2 of Newmann's paper is incomplete, cf. Theorem 4 of this note. 3) Cf. C. Chevalley, loc. cit. 1). 4) In fact for a prime field K with χ(K)=p≠2 there are α, β such that 1+α2+β2=0. Let p=2m+1 and let α1, ... αm be a system of representatives of quadratic rests mod. p. If α2+ β2+1 were_??_0(mod. p) for any, α β then 0, α1, ..., αm, -1 -α1,..., -1-αm (mod. p) would be a complete system of representatatives mod p. If we add them all, then -m≡1+2+...+(p-1) ≡0 (mod. p), which is a contradiction. 5) In fact 1+λ2≡0 (mod. p) has a solution if and only if p≡1 (mod. 4). 6) Cf. L. E. Dickson, Algebren und ihre Zahlentheorie (1927). or R. Brauer-E. Noether, S. B. Preuss. Akad., (1927). See also Appendix I. Cf also A. A. Albert, Structure of algebras, (1939), p. 147, Theorem 27. 7) This follows immediately from a well-known theorem of algebra (for example M. Deuring, Algebren (1934), p. 58, Satz 1). See also Appendix II. 8) See Appendix, III. Cf. also A. A. Albert, loc. cit 6). 9) See Appendix IV. 10) See Appendix V. 11) For the case when K is the complex number field this result is given by M. H. A. Newmann, loc. cit. 1). 12) For the case when Λ is the real number field, this result is given by A. S. Eddington for m=2, and by M. H. A. Newmann for m=3. For m>3 the result of Newmann (Theorem 2) is incomplete. He misses the possibility of the case where all matrices are purely imaginary.