# Algebra Some Current Trends: Proceedings of the 5th National by Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), By Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), Luchezar L. Avramov, Kerope B. Tchakerian (eds.)

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Extra resources for Algebra Some Current Trends: Proceedings of the 5th National School in Algebra held in Varna, Bulgaria, Sept. 24 – Oct. 4, 1986

Sample text

Let rn be the size of the unique indecomposable C(f , ~)-matNx factorization COROLLARY 3 . 1 0 . of f. degD(f,~), ifniseven. m In particuIar, m divides dn-1. Proof. (degD(f,~))2, r . j 2 . ( d e g D ( f , ~))2, if n is odd if n is even. The formula follows. EXAMPLE 3 . 1 1. [] I ~ t f -----Xal+aXa2 E Q[X1, X2], a ~ 0. T h e n the discriminant of f is ( - 1 ) a + l a . Assume t h a t the order of ~ in Q . / Q . d is d. Furthermore, let k :----Q(~), where ~ E C is a d-th primitive root of unity. We will determine a) The unique indecomposable C0(f, ~)-matrix factorization of f .

2S (1985J, 359-374). 4. G. Bergman. Coproducts and some universal ring constructions. Trans. Am. Math. , 200 (1974), 33-88. 5. N. Gerasimov. On free linear groups [in Russian). The 17-th AllUnion Algebraic Conference. Minsk, 1983, part 1, 52-53. 6. N. Gerasimov. Groups of units of free products of rings (in Russian). Mat. , 134, (1987), 42-05. (English translation: Math. USSR, Sb. (to appear)). 7. A. Suslin. On the structure of the special linear group over polynomial rings (in Russian). Izv.

T } w e s e t g(t) = ~ = , (i~-1) C C , and g ( t ) = (\$'1 . . , ~ ' t - - l , ~ + 1,\$t+1 . . . ~t) e G . After having ordered the elements of G lexicographically with respect to the smallest positive remainder we define square matrices %~,92 of size d~ for all gl, g2 E G by setting if gl = g ~ andg2 =g~ otherwise. I (%l,g2)g~,g~ := 0 Then 4: Md(C) @ ... (~ ~ ••• ~ eixj1 Ma(C) eitj t --* M~t(C) ~ Egl,g 2 where gl = (Zl. . . ~t) and g2 = (]1 . . . it), is an isomorphism. or). x. • r=2 gEG Next we want to give an inductive description of ~,~.