By H. Lausch, W. Nobauer

ISBN-10: 0444104410

ISBN-13: 9780444104410

ISBN-10: 0720424550

ISBN-13: 9780720424553

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**Example text**

We have I F1(A) 1 = I A =- I A I. It suffices to show that J PI( A )I s I A / . Let W = W ( A U { x } ) be the word algebra over 9, and w E W . The number of elements of DU AU { x } involved in w, each element counted with the multiplicity it appears in w, Will be called the length of w. Being of the same length is an equivalence relation on W. Let C , be the class consisting of all words of length n. C , is a subset of ( Q U A U {x}>", hence IC,] =s I(Q'JAU{x})"J= J Q U A U { x } l "= I Q U A U { x } l = (QI+ 38 POLYNOMIALS AND POLYNOMIAL FUNCTIONS CH.

C,). c) a, # 1, co = 1. We apply a similar argument as in b). d) a, = 1, co = 1. Then ulu2 = (ao,n,, a,, . ,n,+p,, I), for n, f -pl, and uluz = (ao,n,, a,, . , n,-,, a,-,), for n, = -pl. Furthermore uZu3 = (I, pl+ql, cl, .. ,q,, c,) if q1 F -pl, and u2u3 = (~1,q 2 , czy . CJ if q1 = -pl. Thus (u1u2)u3is obtained from (ao,n,, . , n,+pl, 1, ql, . ,q,, c,) by reduction, for n, # -pl, and from (ao,n,, . ,ar-13 q1, c,, . , q,, c,), for n, = -pl. u , ( u ~ u ~is) obtained from (ao,n,, . , nrr Lp,+q,, c1, .

Bk)be a solution of the system in some %-extencion B of A. 21, that f(b,, . , bk) = g(b,, . , bk), hence, for a, b E A , a0b implies a = 6. Thus 0 is separating. 3. Theorem. Let p , = q,, i E I , be any algebraic system over ( A , %) in X = {xl,. Then one of the ,following statements is true: a) The system has at most one solzrtion in every 8-extension of A . b) For e i w y cardinal u, there exists some %-extension C of A such that the set oj all solutions oj the system in C has cardinality greater than U.

### Algebra of Polynomials by H. Lausch, W. Nobauer

by Ronald

4.1