By H. Lausch, W. Nobauer

ISBN-10: 0444104410

ISBN-13: 9780444104410

ISBN-10: 0720424550

ISBN-13: 9780720424553

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Extra resources for Algebra of Polynomials

Sample text

Whereas polynomial functions over commutative rings with identity, and even more in the case of fields, have been considered in algebra and analysis from time immemorial, the concept of a polynomial has 44 POLYNOMIALS AND POLYNOMIAL FUNCTIONS CH. 1 not become relevant until the rise of modern algebra. 11, that means by using certain sequences. It should be mentioned that a similar construction performed with all sequences (uo, ul, . g. K E R T ~ S[I]). Z Some of our definitions for polynomials over commutative rings with identity make sense also for formal power series, and also some of our results have their counterparts for formal power series.

And inserting a factor a, = 1 in front of every factor x;I where v # 0 in these words, we get words vl, v 2 of %I representing the same elements as w l , w,. Hence v1 = v 2 which is a contradiction since w1 # w2 implies v1 # vz. % is therefore a normal form system for G[xl, . . ,Xk-11 [x,] and, by Th. 6, also one for G[x1, . , xk]. 22. Corollary. Let X = {xl, . 3. Then G(X, %), {pl, yz>is a free product of the group G and the free group F(X, 8)in the variety 8 of groups. Proof. 33 we have just to prove that v Z is a monomorphSince F(X, 8)is generated by X,we have v1 = ism.

Thus a, E P,( A ) . 22. Lemma. Let n a 2, e E P, ( A ) , and y E F, ( A ) such that . * > gn), for (g1, y k , , * * * , g n#e(g,, ) - - - > g , )for , (gp v(g1, * * - 9 8,) = dg,, - . - > gn) f (~1,. 0 . -:,u,), for some (ul, . ,u,) E A", then y E Pn(A). proof. Suppose that y(u,, . ,u,) = c, and let a f b E A . 21. Then a,€P,,(A),thus a,(g,, . ,9,) = v,(aj, g,, . , gn), for some word wl. Let x E Fz ( A ) be defined by v, for u z b x(u,v> = c, for u = b. 23. We complete the proof of the theorem for case a).