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Galois Theory, by David A. Cox
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An introduction to one of the most celebrated theories of mathematics
Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox’s Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics.
Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel’s theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami. Anyone fascinated by abstract algebra will find careful discussions of such topics as:
- The contributions of Lagrange, Galois, and Kronecker
- How to compute Galois groups
- Galois’s results about irreducible polynomials of prime or prime-squared degree
- Abel’s theorem about geometric constructions on the lemniscate
With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike.
- Sales Rank: #2325762 in Books
- Published on: 2004-09-21
- Original language: English
- Number of items: 1
- Dimensions: 9.49" h x 1.52" w x 6.48" l, 2.07 pounds
- Binding: Hardcover
- 584 pages
Review
"This book provides a very detailed and comprehensive presentation of the theory and applications of Galois theory." (Mathematical Reviews, Issue 2006a)
"Happily, Cox's book reads more like a monograph, making a solid case for new subjects rather than rapidly treating a classical one." (CHOICE, September 2005)
" … offers a careful discussion … and will certainly fascinate anyone interested in abstract algebra: a remarkable book!" (Monatshefte fur Mathematik, August 2006)
From the Back Cover
An introduction to one of the most celebrated theories of mathematics
Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox’s Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics.
Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel’s theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami. Anyone fascinated by abstract algebra will find careful discussions of such topics as:
- The contributions of Lagrange, Galois, and Kronecker
- How to compute Galois groups
- Galois’s results about irreducible polynomials of prime or prime-squared degree
- Abel’s theorem about geometric constructions on the lemniscate
With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike.
About the Author
DAVID A. COX is a professor of mathematics at Amherst College. He pursued his undergraduate studies at Rice University and earned his PhD from Princeton in 1975. The main focus of his research is algebraic geometry, though he also has interests in number theory and the history of mathematics. He is the author of Primes of the Form x2 + ny2, published by Wiley, as well as books on computational algebraic geometry and mirror symmetry.
Most helpful customer reviews
17 of 19 people found the following review helpful.
Very nice book
By Julio Cesar De Yncera
This is a wonderful book.
One of the things about abstract algebra is that for the non initiate you tent to loose sight of the problems that motivated an original concept. This book goes an explains the history behind every step. Even some of the demonstrations contain references like (this step here is following a demonstration given by Gauss or Lagrange etc) it is very interesting reading.
Similar to Galois Theory, Third Edition by Ian Stewart but this book containts a lot more detail.
there is a lot of reference to classic works by Galois,Gauss,Cauchy
12 of 13 people found the following review helpful.
Decent textbook with some good extras
By Viktor Blasjo
Part 1 (pp. 1-70) deals with some classical algebra (the cubic equation, symmetric polynomials, the fundamental theorem of algebra). It is nice to see the classical roots emphasised, but I think this could have been done in a much more structured and efficient manner. The chapter on the cubic is 20 pages long and involve 28 exercises. One is impatient already---where's the freakin' Galois theory? Actually, there is still another hundred pages until the definition of the Galois group.
Part 2 (pp. 71-188) develops the Galois theory. Modern Galos theory is couched in the language of field extensions (chapters 4-5); more precisely, the Galois group of a field extension is the group of automorphisms that keeps the base field fixed (chapter 6), and the key to the theory is the "Galois correspondence" between the structure of this group and structure of the field extension (chapter 7).
Part 3 (pp. 189-309) deals with applications. The standard applications are here of course (solvability by radicals; straightedge-and-compass constructions; finite fields and their polynomials) but there are also some more novel ones: automorphisms in geometry (finite subgroups of linear fractional transformations); the "casus irreducibilis" (it is not always possible to express real roots by real radicals); Gauss's work on roots of unity (Gauss showed the solvability by radicals of x^p-1=0 by constructing radical expressions for primitive roots of the intermediate field extensions); "origami" (constructions using straightedge, compass and paper folding).
Part 4 (pp. 310-508), "Further Topics", is what sets this book apart from the usual books. In chapter 12 we study some early works on Galois theory. Lagrange's work on solvability by radicals was the obituary for purely classical methods, but as so often before the grave site soil proved fertile and from here Galois sprang forth with his brilliant little paper containing virtually all the ideas we have seen so far. But Galois's insights into solvability by radicals go beyond the insolvability of the quintic, as we see in chapter 14. In fact, his paper culminates with the theorem that if f is of degree p and f=0 is solvable by radicals then, for any two roots a,b of f, Q(a,b) is the splitting field of f. Going further calls for more sophisticated group theory--we study the case when f is of degree p^2, alluded to by Galois, which is about understanding the solvable primitive subgroups of S_(p^2). Chapter 13 treats methods for computing Galois groups. Finally, Cox has saved the best for last: chapter 15 is on Abel's suggestive work on the division of the lemniscate by ruler and compass (n-division is possible when n is a product of a power of 2 by distinct Fermat primes, just as in the case of the circle). Abel had the idea to employ an analog of the sine function, which is given as the inverse of an arc length integral. This function is not only periodic like the sine but doubly periodic in the complex plane, and it has not only addition formulas but formulas for complex multiplication, which we use in our proof of Abel's theorem. Throughout the book one has grown sick and tired of Cox's abusive use of exercises -- arguments are often shortened by statements like "in exercise x you will show so-and-so; therefore ...". Cox has made sure to end on a high note in this respect: after much preparation the proof of Abel's theorem is just over two pages, but it contains no less than eight references to exercises.
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