id: 06255292
dt: b
an: 2015b.00740
au: Finston, David R.; Morandi, Patrick J.
ti: Abstract algebra. Structure and application.
so: Springer Undergraduate Texts in Mathematics and Technology. Cham: Springer
(ISBN 978-3-319-04497-2/hbk; 978-3-319-04498-9/ebook). ix, 187~p.
(2014).
py: 2014
pu: Cham: Springer
la: EN
cc: H45 F65 H65 P25
ut: identification numbers; error correcting codes; ring; field; group;
cryptography; geometric pattern
ci:
li: doi:10.1007/978-3-319-04498-9
ab: This book can be used in various situations: for an applied algebra course,
for courses designed for secondary mathematics teachers, for programs
of Master’ degree in middle school mathematics education or a Master
of Arts in Teaching Mathematics. The authors try “to develop interest
in abstract algebra by introducing each new structure and topic via a
real-world application". The principal aspects presented here are:
identification numbers, modular arithmetic, linear algebra over finite
fields, isometries of real plane or the ruler and compass
constructions. The theoretic aspects are accompanied by numerous
exercises, even the “proofs of a few propositions being left as
exercises because they give opportunities to employ important
techniques that have been used earlier and will arise again. Some of
the exercises refer to electronic supplementary materials (EMS) in the
form of MAPLE worksheets". The book is structured in ten chapters. The
first chapter investigates the mathematics of identification numbers
and modular arithmetic. There are many types of identification numbers
in common use today: the United States Postal Service zip code, the
Universal Product Code (UPC) used for consumer products and the
International Standard Book Number (ISBN). The modular arithmetic is
used in order to investigate the error detection capabilities of the
various identification number schemes enumerated above. Then, some
error correcting codes are discussed (Hamming code or Golay code). This
concepts enable methods to encoding and transmit information in ways
that allow both direction (recording or transmitting) and correction of
errors. The next three chapters present the theory of rings and fields,
some aspects of linear algebra (vector spaces, linear independence,
spanning, bases, including linear codes) or quotient rings and fields
extensions. These last notions have direct implication in what is
called ruler and compass construction, presented in Chapter 6. Here,
some steps in proofs are given as exercises within the text, especially
because they are fun. Chapter 7 introduces the notion of cyclic code,
constructed from quotient rings of $\mathbb{Z}_{2}[x]$. One advantage
of this construction is that it can guarantee a certain degree of error
construction. The final two applications of abstract algebra discussed
are to cryptography (i.e., secure transmission of private information)
and to the classification of geometric patterns in the plane
$\mathbb{R}^{2}$. The algebraic structure at the heart of both
applications is that of a group. So, in the Chapters 8 and 9 are
introduced the elements of group theory necessary for the most common
contemporary method to encrypt electronic passwords.
rv: Florentina Chirteş (Craiova)