Book description
Explore the foundations and modern applications of Galois theory
Galois theory is widely regarded as one of the most elegant areas of
mathematics. A Classical Introduction to Galois Theory develops
the topic from a historical perspective, with an emphasis on the
solvability of polynomials by radicals. The book provides a gradual
transition from the computational methods typical of early literature
on the subject to the more abstract approach that characterizes most
contemporary expositions.
The author provides an easily-accessible presentation of fundamental
notions such as roots of unity, minimal polynomials, primitive
elements, radical extensions, fixed fields, groups of automorphisms,
and solvable series. As a result, their role in modern treatments of
Galois theory is clearly illuminated for readers. Classical theorems
by Abel, Galois, Gauss, Kronecker, Lagrange, and Ruffini are
presented, and the power of Galois theory as both a theoretical and
computational tool is illustrated through:
- A study of the solvability of polynomials of prime degree
- Development of the theory of periods of roots of unity
- Derivation of the classical formulas for solving general
quadratic, cubic, and quartic polynomials by radicals
Throughout the book, key theorems are proved in two ways, once using
a classical approach and then again utilizing modern methods. Numerous
worked examples showcase the discussed techniques, and background
material on groups and fields is provided, supplying readers with a
self-contained discussion of the topic.
A Classical Introduction to Galois Theory is an excellent
resource for courses on abstract algebra at the upper-undergraduate
level. The book is also appealing to anyone interested in
understanding the origins of Galois theory, why it was created, and
how it has evolved into the discipline it is today.
STEPHEN C. NEWMAN, MD, MSc, is Professor Emeritus of Psychiatry at
the University of Alberta, Canada. He has published widely in
psychiatric epidemiology and epidemiologic methods. Dr. Newman is the
author of Biostatistical Methods in Epidemiology (Wiley).
