Download PDF by Lars Garding, Torbjörn Tambour: Algebra for Computer Science

By Lars Garding, Torbjörn Tambour

ISBN-10: 038796780X

ISBN-13: 9780387967806

ISBN-10: 1461387973

ISBN-13: 9781461387978

The goal of this publication is to educate the reader the themes in algebra that are important within the research of laptop technology. In a transparent, concise kind, the writer current the fundamental algebraic constructions, and their functions to such themes because the finite Fourier rework, coding, complexity, and automata thought. The e-book is additionally learn profitably as a direction in utilized algebra for arithmetic students.

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2) if (a, N) > 1 end. 3) if (a, N) = 1 and (1) does not hold, end. 4) go to 1). By the theorem and the note after it, the chance of N being composite when the algorithm has not stopped after one step is at most 1/2. Hence, if the algorithm has not stopped after n steps, the chance that N is composite is at most 2- n . The practical value of the test is of course bound by the cost of computing (a, N) and the two sides of (1). Since Euclid's algorithm for the pair N, a has at most O(log N) steps, (see the exercises p.

Let A = Za1 + ... + Zak ~ A'. Then A is free. If a is in A', then there are integers n '" 0, n1, ... ,nk such that na + n1 a1 + ... + nkak = O. ), there is an integer m '" 0 such that ma E A for all a E A', or mA' ~ A. But A' is torsion-free, so the map a -+ ma is injective, and A' is isomorphic to a submodule of the free module A. Hence A' is free by the lemma. Define F = Za1 + ... + Zak. = = Then F is free. In fact, if n1a1 + ... + nkak 0, then n1a1 + ... + nkak 0 and n1 = ... = nk = O. We claim that A is the direct sum of F and T(A).

4 The structure of finite modules This section, which is more than a terminological exercise, will show how finite modules are built from cyclic ones. Changed to multiplicative notation, the results also apply to commutative groups. We know that the order of any of its elements divides the order of a finite cyclic module. For use later we shall now prove a stronger statement. First LEMMA. Let a and b be elements of orders m and n in a module M. If m and n are coprime, the order of a + b is mn. If n does not divide m, then the module Za + Zb has elements of order> m.

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Algebra for Computer Science by Lars Garding, Torbjörn Tambour

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