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Download e-book for kindle: Algebra of Polynomials by H. Lausch, W. Nobauer

By H. Lausch, W. Nobauer

ISBN-10: 0444104410

ISBN-13: 9780444104410

ISBN-10: 0720424550

ISBN-13: 9780720424553

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Xk]. We have to find, in a finite number of steps, a word of 112 representing p. By Th. 6, there is an isomorphism p : D[x,, . , x k ] D[x,, . , xk-J [xk]fixing DU {xl, . , xk}elementwise. By Th. 1I , we can find a representation p = (a,n xk)U b, where u,, b, 6D[x,, . ,xk-,], a, == b,, in a finite number of steps. By induction, in a finite number of steps, we can find representations of a,, b, in the 112 belonging to D[xl, . , xk-,]. By substituting these representations into p = (a,nx,)U b, and applying the laws of 23,we get a representation of p by a word of 9.

A,x+a,);= a,,5"+an-,E"-l+ e : R [ x ] S1 + . . +a,~+a,,. Let e : S -,S1 be the embedding isomorphism. Then e-'e(a,x"+ a,,-,X"-l+ ... +a,x+ a,) = (ao,a,, . , a,,, 0, 0, . ). We conclude that different words in %U{O} represent different elements of R [ x ] . 2. Let R again be a commutative ring with identity, considered as a n algebra of 93 and suppose that X = (xl, . R(X, 8)will be denoted by R[x,, . We want to find a normal form system for R[x,, . ,xk], but first of all simplify our notation.

Xk]. Proof. a) We first show for k = 1 that, in a finite number of steps, we can find a word of % representing p E B[x,, . , xk], for every representationp = w ( q , xi). Since a = (anx-1)U(anx) and x = (0nx-l)U (1 nx), we can use induction on the minimal rank of w. By the laws of '8,we have [(an x- 1) u (bnx)l u n x-1) u (dn = [(UUC) nX- 11u [(bud )nXI, [(anx-1) u (bn x)l n [(cnx-l) u ( d n x)] = [(anC) n x-11 u [(bnqnXI, n x- 1) u (bnXI]- 1 = (a- 1ux) n(b- 1ux- 1) = (a-lnb-1)u(a-lnx-l)u(b- in x) = (a-ln x-l) u (b-ln x) u (a-ln b-1 n x-1) u (a-w b - i n x) = (a-lnx-yu (b-lnx).

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Algebra of Polynomials by H. Lausch, W. Nobauer

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