e-book

Algebra: A Computational Introduction

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This text is an introduction to algebra for undergraduates who are interested in
careers which require a strong background in mathematics. It will benefit students
studying computer science and physical sciences, who plan to teach mathematics
in schools, or to work in industry or finance. The book assumes that
the reader has a solid background in linear algebra. For the first 12 chapters elementary
operations, elementary matrices, linear independence and rank are important.
In the second half of the book abstract vector spaces are used. Students
will need to have experience proving results. Some acquaintance with Euclidean
geometry is also desirable. In fact I have found that a course in Euclidean geometry
fits together very well with the algebra in the first 12 chapters. But one can
avoid the geometry in the book by simply omitting chapter 7 and the geometric
parts of chapters 9 and 18.

The material in the book is organized linearly. There are few excursions away
from the main path. The only significant parts which can be omitted are those
just mentioned, the section in chapter 12 on PSL(2, Fp), chapter 13 on abelian
groups and the section in chapter 14 on Berlekamp's algorithm.

The first chapter is meant as an introduction. It discusses congruences and
the integers modulo n. Chapters 3 and 4 introduce permutation groups and linear
groups, preparing for the definition of abstract groups in chapter 5. Chapters 8
and 9 are devoted to group actions. Lagrange's theorem comes in chapter 10 as
an application. The Sylow theorems in chapter 11 are proved following Wielandt
via group actions as well. In chapter 13, row and column reduction of integer
matrices is used to prove the classification theorem for finitely generated abelian
groups. Chapter 14 collects all the results about polynomial rings in one variable
over a field that are needed for Galois theory. I have followed the standard Artin
- van der Waerden approach to Galois theory. But I have tried to show where
it comes from by introducing the Galois group of a polynomial as its symmetry
group, that is the group of permutations of its roots which preserves algebraic
relations among them. Chapters 18, 19, 20 and 21 are applications of Galois
theory. In chapter 20 I have chosen to prove only that the general equation
of degree 5 or greater cannot be solved by taking roots. The correspondence
between radical extensions and solvable Galois groups I have found is often too
sophisticated for undergraduates.

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Informasi Detail

Judul Seri

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No. Panggil

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Penerbit

Toronto : University of Toronto., 2009

Deskripsi Fisik

xii, 405 Hlm.

Bahasa

English

ISBN/ISSN

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Klasifikasi

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Tipe Isi

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Tipe Media

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Tipe Pembawa

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Edisi

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Subjek

MATEMATIKA

Info Detail Spesifik

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Pernyataan Tanggungjawab

agus

Versi lain/terkait

Lampiran Berkas

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