Download e-book for iPad: 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

Show description

Read Online or Download Algebra of Polynomials PDF

Best mathematics books

Jon F. Claerbout's Fundamentals of geophysical data processing (Blackwell PDF

This publication is ready using machine courses for the research of geophysical information to attempt to figure out the structure of the Earth's inside, a strategy which permits a geophysicist to find petroleum and mineral customers. conventional thoughts of information processing are completely mentioned to supply a great beginning at the topic.

Additional info for Algebra of Polynomials

Sample text

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). By definition of P,(A), every element of A is in P,(A).

G , - A g,), for all (gl, . ,g,) E A". Since y E F,-, ( A ) , by induction, we have y E P,-l ( A ) , thus tp(gl, . ,gn-l) = w,(a,, g,, . , g,-l), for some word wl. Also 3: E P 2 ( A ) by hypothesis, thus ~ ( uv), = w2(bj,u, v), for some word w2. Hence q(gl, . , g,) = w2(bj,wl(ai,g17 . e'p = w2(bj,wl(ui, El, . , &-J,En) is in P,(A). This completes the proof of the theorem. -. - 7 * ? 3. Proposition. Let A be I -poljwomially complete, then A is simple. Proof. By way of contradiction, suppose that A is not simple.

Then the results of 5 11 show that the following three cases are possible: a) A is n-polynomially complete for all n. b) A is n-polynomially complete for n = 1, but for no IZ =- 1. c) A is n-polynomially complete for no n. In case a) we say A is polynomially complete, in case b) A is polynomially semicomplete, and in case c) A is polynomially incomplete. For some important varieties, we are going to investigate now in what way the algebras of these varieties distribute over these three cases. We will consider just algebras A such that I A 1 # 1, for j A j = 1 implies that A is polynomially complete for all varieties.

Download PDF sample

Algebra of Polynomials by H. Lausch, W. Nobauer


by Jeff
4.2

Rated 4.61 of 5 – based on 23 votes