By Lars Garding, Torbjörn Tambour

ISBN-10: 038796780X

ISBN-13: 9780387967806

ISBN-10: 1461387973

ISBN-13: 9781461387978

The objective of this ebook is to coach the reader the themes in algebra that are important within the learn of laptop technology. In a transparent, concise type, the writer current the fundamental algebraic constructions, and their purposes to such themes because the finite Fourier remodel, coding, complexity, and automata idea. The publication is also learn profitably as a path in utilized algebra for arithmetic students.

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Here z is any integer, said to represent the class C(z). Addition, subtraction and multiplication in this set are defined by the formulas (1) C(z) ± C(y) = C(z ± y), C(z)C(y) = C(zy). In order to verify that these definitions make sense and satisfy the axioms we first note that z and y belong to the same class C(z) if and only if z - z == 0 (m) and y - z == 0 (m) from which follows that z - y == 0 (m). Now if z == z' (m) and y == y' (m), we know that z ± y == z' ± y' (m) and zy == z'y' (m). Hence the right sides above do not depend on the choice of representatives of the classes C(z) and C(y).

Prove that Pr(75) has the type (20,2) and write the group explicitly as a direct product of two cyclic groups. (Hint. IPr(75)1 = 40 and we know that Pr(75) is not cyclic. ) 2. What is the type of Pr(lOO)? 5 Appendix. 5 Appendix. Finitely generated modules Let A be a module. We say that A is finitely generated if there is a finite set of elements aI, ... ,ak of A such that every a in A can be written for some integers nl, ... ,nk. The elements aI, ... ,ak are called generators of A. If A is finitely generated, we say that it is free (on al,'" ,ak) if When A is a module, we denote by T(A) the submodule of A consisting of all elements a of A such that na = 0 for some integer n i= O.

Let Za be a cyclic module with m elements. Show that Za and ZlmZ are isomorphic. R. Let LeN C M be three modules. Note that NIL can be considered as a submodule of MIL. Show that (MIL)/(NIL) is isomorphic to MIN. (The second homomorphism theorem. Hint. ) The modules Hom(M, N) Let M and N be two modules and f,g: M Define their sum f + g by -+ N two module morphisms. (f + g)(x) = f(x) + g(x). This gives the structure of a module to the set Hom(M, N) of all homomorphisms from M to N. R. Verify this statement.

### Algebra for Computer Science by Lars Garding, Torbjörn Tambour

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