Book Description
This text offers a clear, efficient exposition of Galois Theory with complete proofs and exercises. Topics include: cubic and quartic formulas; Fundamental Theory of Galois Theory; insolvability of the quintic; Galois's Great Theorem (solvability by radicals of a polynomial is equivalent to solvability of its Galois Group); and computation of Galois groups of cubics and quartics. There are appendices on group theory, ruler-compass constructions, and the early history of Galois Theory. This book provides a concise introduction to Galois Theory suitable for first-year graduate students, either as a text for a course or for study outside the classroom. This new edition has been completely rewritten in an attempt to make proofs clearer by providing more details. The book now begins with a short section on symmetry groups of polygons in the plane, for there is an analogy between polygons and their symmetry groups and polynomials and their Galois groups; this analogy can serve as a guide by helping readers organize the various field theoretic definitions and constructions. The exposition has been reorganized so that the discussion of solvability by radicals now appears later and several new theorems not found in the first edition are included (e.g., Casus Irreducibilis).
Customer Reviews:
A good book for a first date with Artin's version of Galois theory.......2007-07-16
I came to know this book in its first edition and I particularly prefer that one to this second. The contents in this new addition has been expanded a bit, new worked out examples has been added and new details has been given to some proofs, but in my opinion the book has lost some of the lucidity that it claims to have in the preface.
Despite this the book continues being one of the best introduction to Galois theory and I recommend it to anyone pursuing this subject, even abstract algebra itself, for the first time, since it gathers all the elemetary material in a succint form.
But if you really want to get a feeling of the beautiful ideas that ocurred to Galois you must go first to his original writings ("read the Masters" as Edwards recommends), and then come to this book to appreciate the value of the generality of Artin's version of Galois theory in terms of field automorphisms.
Good Reference.......2003-03-09
This text serves as a good reference, however, it is not necessarily well-suited for self-learning. The exercise sets need some revamping.
Sweet and Concise.......2002-07-28
I used this book as my course text book in my Galois Theory course in my undergraduate. It is pretty self-contained, so even if you forget some of Group and Fields Theory, you will still find it very readable. The little book guides to the Great Theorem by Galois (f is solvable by radical if and only if Gal(f/Q) is solvable) by breaking it into many nice sections. A drawback of this book is that some proofs at the end are omitted because they are put as execrises! Also, bear in mind that this book focus in Q and R (rational and real numbers) field only. In any case, I still recommend this book because it is sweet and concise, for which I can read it like a novel!
Classical "Artin's" Galois theory.......1997-12-08
From the definition of a conmutative ring to the fundamental theorem to solvability of equations by radicals in 65 pages, 80 theorems and 106 exercises. The exposition, wich follows the now classical tradition of "Artin's" Galois theory, is quite efficient, packing much material in a limited number of pages. The greatest asset of this book is its nice selection of topics, focusing on the fundamental theorem of Galois theory and its application to solbability of equations by radicals, but pausing to make excursions to finite fields or to work out explicitly some iluminating examples. The style is no-nonsense, crisp but nop hurried. The final 40 pages consist of appendices discussing group theory, ruler an compass constructions and old-fashioned Galois theory. This latter appendix deserves special mention, since it is not customary for a textbook of this size and scope to include such a detailed sketch of the historical motivations behind the theory it describes. One can only agree with the author when he wonders how such thoughts occurred to Galois in the late 1820's, and be grateful to him for providing his reader with material for an answer.
Book Description
What do the music of J. S. Bach, the basic forces of nature, Rubik's Cube, and the selection of mates have in common? They are all characterized by certain symmetries. Symmetry is the concept that bridges the gap between science and art, between the world of theoretical physics and the everyday world we see around us. Yet the "language" of symmetry--group theory in mathematics--emerged from a most unlikely source: an equation that couldn't be solved.
Over the millennia, mathematicians solved progressively more difficult algebraic equations until they came to what is known as the quintic equation. For several centuries it resisted solution, until two mathematical prodigies independently discovered that it could not be solved by the usual methods, thereby opening the door to group theory. These young geniuses, a Norwegian named Niels Henrik Abel and a Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the night before his fatal duel (at the age of twenty) scribbling another brief summary of his proof, at one point writing in the margin of his notebook "I have no time."
The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines. In this lively, engaging book, Mario Livio shows in an easily accessible way how group theory explains the symmetry and order of both the natural and the human-made worlds.
Download Description
"What do the music of J. S. Bach, the basic forces of nature, Rubik's Cube, and the selection of mates have in common? They are all characterized by certain symmetries. Symmetry is the concept that bridges the gap between science and art, between the world of theoretical physics and the everyday world we see around us. Yet the ""language"" of symmetry--group theory in mathematics--emerged from a most unlikely source: an equation that couldn't be solved. Over the millennia, mathematicians solved progressively more difficult algebraic equations until they came to what is known as the quintic equation. For several centuries it resisted solution, until two mathematical prodigies independently discovered that it could not be solved by the usual methods, thereby opening the door to group theory. These young geniuses, a Norwegian named Niels Henrik Abel and a Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the night before his fatal duel (at the age of twenty) scribbling another brief summary of his proof, at one point writing in the margin of his notebook ""I have no time."" The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines. In this lively, engaging book, Mario Livio shows in an easily accessible way how group theory explains the symmetry and order of both the natural and the human-made worlds. "
Customer Reviews:
A Fine book with a few permutations.......2007-09-08
If you are not a mathematician (and I am not), but have an interest in the subject, and a working knowledge of some elementary ideas, this is a terrific book. It has the easiest explanation of symmetry/Galois groups, etc., of any of the books I have tried on the topic -- oh sure, it rambles (as the severe critics here say) -- but try and find some other book on the subject that doesn't immediately drop you far beyond your depth. Livio has a knack for very, very clear explanations and great metaphors (permutations and probability are discussed in terms of finding a mate). I recommend it highly, especially if you can get it with one of Ian Stewart's books on the same topic.
"Don't cry, I need all my courage to die at twenty."...Galois.......2007-02-09
When I came across this book,I thumbed through it and the figures that jumped out at me were a collection of things,mainly about mathematics,puzzles and other things that interest me. I graduated in Electrical Engineering nearly 50 years ago,and have had a lifelong interest in Mathematical Recreations and Puzzles of all sorts. Granted most of the Mathematics I studied has long since left me mainly because of lack of use.However,the lore,beauty,mystery and fascination of Mathematics has remained. A lot of the Mathematics discussed in this book falls into what I think of as Theoretical rather than Applied Mathematics;and then there's that whole area of Recreational Mathematics.
I have read all the other reviews here,and basically agree with all of them.Taken together they do a good job of telling what the book is about and the Mathematicians who searched for those elusive solutions.In fact,there is so much that could be covered that it would take many volumes to even only scratch the surface.
I don't know if I really "know" much more about Group Theory and Symmetry than when I started ,but I still found it a fascinating read. Kind of like a 5-day tour of Europe-Been there,done that,but do I "know" Europe?
Like I said,other reviews have pretty well covered the book;so I won't repeat.
However; I would like to point out a couple of things.
In chapter 6,the 15-Puzzle is discussed. This is one of the all time greatest puzzles.It has interested me for years. If you would like to know more about it,I strongly recommend you read "The 15 Puzzle" by Jerry Slocum and Dic Sonnefeld.After you see this book ,you'll probably agree it is one of the world's most interestting puzzles;and what a history and legend it has. I posted a review of it here on Amazon on June 6,2006.
If you haven't noticed ,the information on this book has a section "Inside the Book" and in this section under "text stats" ,it shows this book has a Fog Index of 16.2. A search on the net will show how it is calculated. It takes a sample of text,and by looking at the lengths of sentences,number of multiple syllable words,paragraphs,and so forth comes up with a number that shows how difficult it is to comprehend. 16.2 is a fairly high level; and that combined with the theoretical math concepts;there is lttle wonder tht many would find this a fairly difficult book to read.Of course,I'm referring to the Mathematical concepts as opposed to the Biographical information.
The author must have done a tremendous amount of research in writing this book, and in the extensive Notes and References he provides a huge amount of information for the reader who wishes to pursue anything further
A lively read for a wide audience.......2007-01-07
Symmetry is the topic of Mario Livio's THE EQUATION THAT COULDN'T BE SOLVED: HOW MATHEMATICAL GENIUS DISCOVERED THE LANGUAGE OF SYMMETRY, and will make an involving read for those involved in either science or art. Mathematicians solved algebraic equations until they came to a stop with the quintic equation, which resisted solution until two mathematical geniuses independently discovered it couldn't be solved using the usual methods. This account of 'group theory' explains both the concept of symmetry and the evolution of its foundations, and makes for a lively read for a wide audience from physicists and science majors to students involved in the arts.
Diane C. Donovan
California Bookwatch
very accessible introduction to group theory and it's history.......2006-09-05
The equation that couldn't be solved is about the history of group theory. The stories of two of it's early contributors Abel and Galois is told in detail. In addition the author provides an accessible overview of group theory. The specific equation that couldn't be solved is the quintic, which cannot be factored in general. That means that while there are specific examples of polynomials with a factor of x raised to 5 or greater that can be factored there is no general formula like the quadratic eqauation that can factor all quintic or higher polynomials. Although originally used to study factoring, group theory has evolved to be about many other things including the mathematical concept of symmetry. Symmetry arises in many parts of mathematics and science so it is very imporant. I came away from this book with a knowledge of the history of group theory and a smattering of knowledge about group theory and it's applications. I highly recommend this book to those people, like me, who are interested in mathematics and would like to peek under the surface to see what it is all about.
Sinusoidal.......2006-08-03
As I was reading this book, my interest level ebbed and flowed; it was like that all the way to the end. For the most part, the book held my attention, but often I got the feeling that the author was straying from the symmetry theme a bit too much. So this is very much a book of peaks and valleys (hence the title of my review).
The most fascinating parts of the book for me are certainly the masterfully-written biographies of Abel and Galois, and the author's discussion of the cubic equation and its gradual solution by dal Ferro, Fiore, Tartaglia, and Cardano. The author is totally smitten by the figure of Galois; he is described in wonderful detail, warts and all: the young hot-tempered revolutionary romantic who let politics consume him, largely as a result of the terrible misfortunes he endured in his personal and academic life, the tragic duel that resulted in his death from peritonitis, and the inexhaustible legacy of group theory which he bequeathed to the world.
The discussion about groups is first-rate. Permutation groups are rightly emphasized. I regret that he does not dwell on group theory more. But for the author to include a detailed description of normal subgroups is a mark of his willingness not to underestimate his readers. There is also a clear and concise discussion of the quantum world, relativity, and even string theory. But on the downside, there is the author's tendency to write in a sort of blithe, whimsical, off-hand manner that can grate on the nerves after awhile.
I was struck on occasion by fascinating statements that had never occurred to me before, such as: "All the electrons in the universe are precisely identical in terms of their intrinsic properties; there is no way to distinguish one from the other." And: "The chief reason we can interpret relatively easily observations of galaxies ten billion light years away is that we find that hydrogen atoms there obey precisely the same quantum mechanical laws they obey on Earth." And: "Female orgasm seems to be less about bonding with a great person than about a cold Stone Age evaluation of the mate's genetic endowment." This last revelation may have far-reaching effects, in the amount of mail which our esteemed author has received from female readers!
In summary, this is a very wide-ranging book, perhaps a little too wide-ranging, and not quite as polished and elegantly written as I would have liked. But it is still a fascinating exploration of group theory and its applications for anyone with a curious and open mind.
Product Description
Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book, Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of particular investigations. Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed. Results in Galois theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular $n$-gons are also presented. This book is suitable for undergraduates and beginning graduate students.
Book Description
This is an updated English translation of "Cohomologie Galoisienne", published more than 30 years ago as one of the very first Lecture Notes in Mathematics (LNM 5). It includes a reproduction of an influential paper of R. Steinberg, together with some new material and an expanded bibliography.
Book Description
An introduction to one of the most celebrated theories 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
Customer Reviews:
Decent textbook with some good extras.......2006-03-20
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.
Very nice book.......2005-12-06
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
Book Description
Clearly presented elements of one of the most penetrating concepts in modern mathematics include discussions of fields, vector spaces, homogeneous linear equations, extension fields, polynomials, algebraic elements, as well as sections on solvable groups, permutation groups, solution of equations by radicals, and other concepts. 1966 edition.
Customer Reviews:
the source!.......2004-04-13
This is modern Galois Theory, straight from the horse's mouth! Galois Theory is taught today using field extensions rather than by actually solving polynomials, students also learn to view a field extension as a vector space over the smaller field; both of these things were pioneered by Artin. The book also has short, clear proofs of all the main theorems. The only problem is that there are no problems to work on, so I have to say this is only a good reference for Galois Theory.
Succinct exposition of modern Galois theory by a pioneer........2003-11-13
Emil Artin's short book gets a mention in most texts on
Galois theory. It is very short - only 60 odd pages. Yet
it is a very clear, complete and readable account of the
essential elements of modern Galois theory. It is based
on lectures he gave over 50 years ago but you might think
it was written only yesterday and is comprehensible to
anyone familiar with current abstract algebra terminology.
And the price makes it a bargain. There are no worked
examples, exercises or index here.
just enjoy.......2002-02-19
during reading this cute booklet, you can surely hear the gentle talk of an old math maven.(from the publishing date, the auther was 44 but that's my impression.) with a cup of coffee, stretch those edgy wrinkles of your brain.
Nicely writien, short........2001-07-20
A friend of mine has a maxim: The shorter a math book, the more likely I am to read it. Artin's Galois Theory is certainly that. It is also an example of Artin's wonderful mathematical style. Gian-Carlo Rota, who took classes from Artin when he was at princeton, said that Artin's proofs were perfect, as though he had gone through all the available proofs to find *the* proof and that was the one he used. Rota felt that this left the student at a disadvantage in that he didn't know about the effort that went into the proof, nor why it is beautiful. I disagree: the proofs in Galois Theory have a certain indescribable beauty to them which left me awestruck at their simplicity. They seem to have all the requisite attributes (as laid out in Hardy's A Mathematician's apology) to be considered beautiful. These notes are by no means complete, but I would suggest them as a suplement to another treatment of field theory (for example, Dummit and Foote or Morandi even though they were based upon Artin's treatment).
Okay if you are interested in matehmatical "classics"........2001-05-14
I agree, to some extent, with the recent two reviewers: Nobody can deny that Emil Artin was a great mathematician, having done a very good job in algebra. That does not necessarily mean his textbooks should be praised *ad infinitum*. I understand some classics remain valuable for an incredibly long period of time ("Morse theory" by Milnor is one of such landmarks that comes into my mind), but I feel scheptical if this one deserves that claim. This book is okay if you are interested in his writing style of many years ago, but not quite so if your main concern is to study Galois theory (or algebra: that makes no difference for that matter) efficiently and effectively. In that case you should turn to more modern textbooks like Cohn ("Algebra" published by Wiley.)
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Field Theory (Graduate Texts in Mathematics)
Steven Roman
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ASIN: 0387276777 |
Book Description
Intended for graduate courses or for independent study, this book presents the basic theory of fields. The first part begins with a discussion of polynomials over a ring, the division algorithm, irreducibility, field extensions, and embeddings. The second part is devoted to Galois theory. The third part of the book treats the theory of binomials. The book concludes with a chapter on families of binomials – the Kummer theory.
This new edition has been completely rewritten in order to improve the pedagogy and to make the text more accessible to graduate students. The exercises have also been improved and a new chapter on ordered fields has been included.
About the first edition:
" ...the author has gotten across many important ideas and results. This book should not only work well as a textbook for a beginning graduate course in field theory, but also for a student who wishes to take a field theory course as independent study."
-J.N. Mordeson, Zentralblatt
"The book is written in a clear and explanatory style. It contains over 235 exercises which provide a challenge to the reader. The book is recommended for a graduate course in field theory as well as for independent study."
- T. Albu, MathSciNet
Book Description
The pioneering work of Abel and Galois in the early nineteenth century demonstrated that the long-standing quest for a solution of quintic equations by radicals was fruitless: no formula can be found. The techniques they used were, in the end, more important than the resolution of a somewhat esoteric problem, for they were the genesis of modern abstract algebra.
This book provides a gentle introduction to Galois theory suitable for third- and fourth-year undergraduates and beginning graduates. The approach is unashamedly unhistorical: it uses the language and techniques of abstract algebra to express complex arguments in contemporary terms. Thus the insolubility of the quintic by radicals is linked to the fact that the alternating group of degree 5 is simple - which is assuredly not the way Galois would have expressed the connection.
Topics covered include:
rings and fields
integral domains and polynomials
field extensions and splitting fields
applications to geometry
finite fields
the Galois group
equations
Group theory features in many of the arguments, and is fully explained in the text. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided.
Customer Reviews:
Fields and Galois Theory.......2006-02-25
This is a short but very good introductory book on abstract algebra, with emphasis on Galois Theory. Very little background in mathematics is required, so that the potential audience for this book range from undergraduate and graduate students, researchers, computer professionals, and the math enthusiasts.
Book Description
This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules.
"In all three parts of this book the author lives up to his reputation as a first-rate mathematical stylist. Throughout the work the clarity and precision of the presentation is not only a source of constant pleasure but will enable the neophyte to master the material here presented with dispatch and ease."—A. Rosenberg, Mathematical Reviews
Customer Reviews:
Pretty good.......2002-11-13
This book is an advanced treatment of field theory and Galois theory and is meant for those readers who have a substantial background in graduate algebra. The subject matter used to be thought of as purely mathematical, but due to the influence of the field of cryptography, it now has many applications. I only read part 1 of the book, so my review will be confined to this part.
The author begins the discussion with field extensions. One can view a field L containing another field K as a vector space over K, and the dimension of L (as a vector space) is then called the 'dimension' of L over K. If one considers a subfield K of a field M, and an additional element u in M, then there is a smallest subfield of M containing K and u. Calling this field K(u), u can be either transcendental or algebraic over K. The author then proves some elementary properties of the field K(u), showing the existence of an irreducible polynomial for u over K. This then motivates him to call a field L containing K 'algebraic' over K if every element of L is algebraic over K. Otherwise L is called 'transcendental' over K. The dimension of K(u) over K is called the degree of u over K. Finding the degree of u can be done by finding the irreducible polynomial for u. The author also proves the arithmetic relation between the dimensions of towers of fields, and this allows him to prove the famous results on the impossibility of ruler and compass constructions. For a field L that lies between fields K and M the author studies the 'stability' of L over K, meaning that every automorphism of M/K sends L into itself. The correspondence between stable fields and normal subgroups of the Galois group of M/K is proven. Splitting fields are introduced as devices to obtain fields that are normal over a given field. A criterion for a splitting field that does not involve polynomials is proven, and the author gives tools that deal with fields of non-zero characteristic, these tools motivating the definition of separability. Splitting fields are normal in characteristic 0, but one must add separability for the same to hold in characteristic p. The unsolvability of the quintic is shown via a discussion on radical extensions of fields. For a field K of characteristic 0, and for a field L lying between K and another field M, where M is a radical extension of K, the author proves in detail that the Galois group of L/K will be solvable. Then if one has a polynomial with coefficients in K, then the Galois group of this polynomial is defined to be the Galois group of a splitting field of the polynomial over K. The Galois group of the polynomial is thought of as a group of permutations of the roots of the polynomial. The author then proves that if K has characteristic 0 and L is a radical extension of K which contains a root of the polynomial, then the Galois group of the polynomial over K is solvable.
Those readers involved in cryptography will find a discussion of finite fields in Part 1. The author's goal is to find the finite fields and determine their structure. He first proves that every nth power of a prime number p will yield a field with p^n elements. The author shows that the Galois theory of finite fields is simple by proving that if K is a finite field contained in another finite field L, then L is normal over K and the Galois group of L/K is cyclic.
The author also shows how the Galois group of an equation can be found explicitly for the cubic and quartic equations. He shows first that for the Galois group of a separable irreducible cubic over a field K is either the alternating group A(3) or the symmetric group S(3). If the characteristic of K is not equal to 2, then it is A(3) if and only if the discriminant is a square in K. For a separable irreducible quartic over K, then for the degree over K of the splitting field of the resolvent cubic of this polynomial, the Galois group is S(4) if the degree is 6, A(4) if the degree is 3, V (a particular normal subgroup of S(4)) if the degree is 1, and either the group of order 8 or cyclic of order 4 if the degree is 2.
Also in part 1, the author studies the reducibility of an equation of the form x^n -a over an arbitrary field. He addresses this reducibility by first proving that one only need be concerned for the case where n is a prime power. Then if p is prime, and "a " does not have any pth root in the field K, then if the prime is odd, then the equation is irreducible over K for any n. If p = 2 and the characteristic of K is 2, then the equation is irreducible over K for any n. If p = 2, n is greater than or equal to 2, and the characteristic of K is not 2, then the equation is irreducible over K if and only if -4a is not a fourth power in K. The author also proves the fundamental theorem of algebra using Galois theory. He does this by first showing that if every extension of K has degree divisible by a prime p, then every extension of K has degree a power of p.
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Galois Theory
Jean-Pierre Escofier
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Similar Items:
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Galois Theory (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)
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Field Theory (Graduate Texts in Mathematics)
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Galois Theory (Graduate Texts in Mathematics)
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Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics)
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A Course in Galois Theory
ASIN: 0387987657 |
Book Description
The new book on basic Galois theory by Jean-Pierre Escofier is refreshingly different from the many books on the same subject already on the market. The book covers the standard basic material -- symmetric polynomials, field extensions, normal and Galois extensions, the Galois correspondence, cyclotomic extensions, solvable groups, finite fields and separable and non-separable extensions. However, it also contains the following original features: a sketch of the early history of the subject from a very human viewpoint, containing a large number of excerpts from original works and a discussion of the problems of notation, discovery, mathematical habits and disputes of former times, a complete chapter on explicit constructions with ruler and compass, a chapter on the life of Evariste Galois and a chapter on recent developments which attempts to give an idea of what researchers in Galois theory are working on today. The book requires only standard algebra (groups, rings, fields) as background, and is eminently suitable for an undergraduate text on Galois theory. It contains a large number of useful exercises, mostly with their solutions. Because of its exciting and very human approach, it should attract even students who are not mathematics majors to the beautiful subject of Galois theory.
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