id: 05859358
dt: b
an: 2011c.00606
au: Carstensen, Celine; Fine, Benjamin; Rosenberger, Gerhard
ti: Abstract algebra. Applications to Galois theory, algebraic geometry and
cryptography.
so: Sigma Series in Pure Mathematics 11; De Gruyter Graduate. Berlin: Walter de
Gruyter; Lemgo: Heldermann Verlag (ISBN 978-3-11-025008-4/pbk). xi,
366~p. (2011).
py: 2011
pu: Berlin: Walter de Gruyter; Lemgo: Heldermann Verlag
la: EN
cc: H45 H75 P20
ut: groups; rings; fields; subgroups; ideals; polynomials; fields extensions;
solvability; modules; cryptography
ci:
li: doi:10.1515/9783110250091
ab: This book is an introductory text on abstract algebra and the authors
assume that the readers are familiar with Calculus and with some linear
algebra, primarily matrix algebra and the basic concepts of vector
spaces, bases and dimensions. All other necessary material is
introduced and explained in the book. The material is presented
sequentially so that the polynomials and field extensions precede an
in-depth look at group theory. The centerpiece of these notes is the
development of Galois theory and its important applications, especially
the insolvability of the quintic. The basic algebraic structures
(group, rings and fields) are introduced in the first three chapters.
The first notion is briefly exposed but the last two notions are
presented here in detail, so that we can find enough information
regarding factor rings and ring homomorphisms, quotient fields,
subrings and ideals (prime, maximal or principal), integral domains or
principal ideal domains. In Chapter 3, the fundamental theorem of
arithmetic is revised (its proof and several other ideas from classical
number theory) and it is proved that there are many other integral
domains where this also holds (called unique factorization domains).
Then the authors begin the theory of polynomials and polynomial
equations over rings and fields (including the fundamental theorem of
algebra or of symmetric polynomials), develop the main ideas of field
extensions and adjoining elements to fields. Regarding these, they
translate three problems of constructions using a straightedge and
compass into the language of field extensions and prove that each of
these problems is insolvable in general and give the complete solution
to the construction of the regular $n$-gons. The authors finish the
first part of the book with splitting fields and normal extensions,
presenting another proof for the fundamental theorem of algebra. The
concept of a splitting field is essential to the Galois theory of
equations. After this, the necessary material from group theory needed
to complete both the insolvability of the quintic and solvability by
radicals in general is presented. Hence the middle part of the book,
Chapters 9 through 14, are concerned with group theory, including
permutation groups, solvable groups, abelian groups and group actions.
Chapter 14 is somewhat off to the side of the main theme of the book.
Here the authors give a brief introduction to free groups, group
presentations and combinatorial group theory. Finally, after all of
these presentations, the last but most important part of the book
begins. They study general normal and separable extensions and the
fundamental theorem of Galois theory. Using this, the authors present
several major applications of the theory, including solvability by
radicals and the insolvability of the quintic, the fundamental theorem
of algebra, the constructions of regular $n$-gons and the famous
impossibilities: squaring the circle, doubling the cube and trisecting
an angle. Some aspects from the theory of modules follow (vector spaces
being crucial in the study of fields and Galois theory since every
field extension is a vector space over any subfield), finitely
generated abelian groups, integral and transcendental extensions
(including the transcendence of $e$ and $π$) and algebraic geometry
(this involving the study of algebraic curves which roughly are the
sets of zeros of a polynomial or of a set of polynomials in several
variables over a field). The book is finished in a slightly different
direction giving an introduction to algebraic and group based
cryptography.
rv: Florentina Chirteş (Craiova)