Galois Theory (Universitext)

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.

Galois Theory: Lectures Delivered at the University of Notre Dame (Notre Dame Mathematical Lectures, Number 2)

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the source!
Succinct exposition of modern Galois theory by a pioneer.
just enjoy
Nicely writien, short.
Okay if you are interested in matehmatical "classics".

Galois Theory: Lectures Delivered at the University of Notre Dame (Notre Dame Mathematical Lectures, Number 2)
Emil Artin , and Arthur N. Milgram
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486623424

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.)

Field Theory and Its Classical Problems (Carus Mathematical Monographs ; No. 19) (Carus Mathematical Monographs)

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Some good classical algebra
Outstanding! One of my favourites!
An excellent introduction to the field (pun intended)

Field Theory and Its Classical Problems (Carus Mathematical Monographs ; No. 19) (Carus Mathematical Monographs)
Charles Robert Hadlock
Manufacturer: The Mathematical Association of America
ProductGroup: Book
Binding: Paperback

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ASIN: 088385032X

Book Description

Field Theory and its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the construction of regular n-gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals and beyond. The logical pathway is historic, but the terminology is consistent with modern treatments. No previous knowledge of algebra is assumed. Notable topics treated along this route include the transcendence of e and p, cyclotomic polynomials, polynomials over the integers, Hilbert’s irreducibility theorem, and many other gems in classical mathematics. Historical and bibliographical notes complement the text, and complete solutions are provided to all problems.

Customer Reviews:

Some good classical algebra.......2006-12-22

First we study the classical construction problems. With ruler and compass we can construct precisely the numbers built up by rational operations and square roots, since our construction tools are "limited to degree two". For some specific number it could be very easy to prove that it is not constructible. This is the case for the cube root of 2 and the cosine of 20---the constructions equivalent to the classical problems of doubling the cube and trisecting an angle. These proofs are based on first proving that if we were trying to reach these number from the rationals then one more square root could never be the final step, so if they were constructible they must have been among the rationals in the first place, and since they are clearly irrational it follows that they are not constructible. But this approach does not work for proving that it is impossible to construct pi (square the circle). Here a great detour is required, using the "upper bound" on constructibility that any constructible number is algebraic. The proof that pi is not algebraic doesn't have anything to do with field theory; instead, for example, we are asked to work out a chunk of complex function theory in the exercises (complex exponential function, fundamental theorem of calculus). The next chapter follows a similar pattern: field theory is introduced in all its glory, and we wish to use it to study construbtibility of n-gons. In accordance with the above, "constructible" is now equivalent to "obtainable by field extensions of degree two", and this allows us to rule out many n-gons as impossible. But this fancy-pants theory cannot help us prove that the 2^n Fermat prime n-gons are in fact constructible---here we cannot improve on the ingeniously contrived proof of Gauss. So far, then, "field theory" is a bit of a failure since it had nothing to do with the proofs of our two most interesting theorems---indeed, its pointlessness forces Hadlock to resort to two "it is natural to ask..." proclamations in two pages (pp. 72-73). But we soon see that field theory is not entirely useless when we study the classical Galois theory of solvability by radicals. This is a standard treatment just as in any Galois theory textbook, but we then go deeper into follow-up questions: unsolvability by radicals of the general quintic of course follows from the unsolvability of its Galois group S_n, but then we set out to find specific polynomial equations that are not solvable by radicals. Some hard work with classical algebra and analysis enables us to prove Hilbert's irreducibility theorem: for any irreducible polynomial in n+1 variables we can pick rational values for n of the variables so that the resulting polynomial in one variable is still irreducible. With this result we can construct a polynomial of degree n whose Galois group has order at least n!, so it must equal S_n, so all roots are interchangeable, so all roots belong to the same factorisation. It follows that there are equations of any degree greater than 4 with no roots expressible in terms of radicals and that there are unconstructible numbers of degree 2^m (not surprising since the general solution of the quartic is based on reduction to a cubic and so involves cube roots).

Outstanding! One of my favourites!.......2004-04-12

This book gives a very concrete intro to Galois theory & field theory & would be an excellent supplement to an advanced course on algebra. It has some group theory in it but its emphasis is on field theory because its focus is ONLY on Galois theory & its applications, so you'll have to find another book on general abstract algebra. But for Galois theory this text really helped me out with its fairly concrete treatment. Look at this to get a more tangible version of more abstract stuff. Each chapter has a bibiolgraphy at the end of it & each one is a goldmine for interesting stuff about abstract algebra, history, numbers, etc.(...) One could also think of this book as a jumping-off point to all kinds of other stuff it scratches the surface of.

An excellent introduction to the field (pun intended).......2003-02-08

Hadlock writes in a brisk, thorough style, leaving no stone unturned in what I believe to be an excellent introduction to the subject. I bought this book as an undergraduate student, upon completion of the first course in modern algebra, and it has proved to be an indispensable tool since.

Topics in Galois Theory (Research Notes in Mathematics, Volume 1) (Research Notes in Mathematics)

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Topics in Galois Theory (Research Notes in Mathematics, Volume 1) (Research Notes in Mathematics)
Jean Pierre Serre
Manufacturer: Jones & Bartlett Publishers
ProductGroup: Book
Binding: Paperback

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ASIN: 0867202106

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Written by one of the major contributors to the field, this book is packed with examples, exercises, and open problems for further edification on this intriguing topic.

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Groups, Rings and Galois Theory
Victor P. Snaith
Manufacturer: World Scientific Publishing Company
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Binding: Paperback

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ASIN: 9812386009

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This book is ideally suited for a two-term undergraduate algebra course culminating in a discussion on Galois theory. It provides an introduction to group theory and ring theory en route. In addition, there is a chapter on groups — including applications to error-correcting codes and to solving Rubik's cube. The concise style of the book will facilitate student-instructor discussion, as will the selection of exercises with various levels of difficulty. For the second edition, two chapters on modules over principal ideal domains and Dedekind domains have been added, which are suitable for an advanced undergraduate reading course or a first-year graduate course.

Groups as Galois Groups: An Introduction (Cambridge Studies in Advanced Mathematics)

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Groups as Galois Groups: An Introduction (Cambridge Studies in Advanced Mathematics)
Helmut Volklein
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521562805

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This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realization of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.

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Galois Theory (Universitext)
Steven H. Weintraub
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 0387287256

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Classical Galois theory is a subject generally acknowledged to be one of the most central and beautiful areas in pure mathematics. This text develops the subject systematically and from the beginning, requiring of the reader only basic facts about polynomials and a good knowledge of linear algebra. Key topics and features of this book: Approaches Galois theory from the linear algebra point of view, following Artin; Develops the basic concepts and theorems of Galois theory, including algebraic, normal, separable, and Galois extensions, and the Fundamental Theorem of Galois Theory; Presents a number of applications of Galois theory, including symmetric functions, finite fields, cyclotomic fields, algebraic number fields, solvability of equations by radicals, and the impossibility of solution of the three geometric problems of Greek antiquity; Provides excellent motivaton and examples throughout. The book discusses Galois theory in considerable generality, treating fields of characteristic zero and of positive characteristic with consideration of both separable and inseparable extensions, but with a particular emphasis on algebraic extensions of the field of rational numbers. While most of the book is concerned with finite extensions, it concludes with a discussion of the algebraic closure and of infinite Galois extensions. Steven H. Weintraub is Professor and Chair of the Department of Mathematics at Lehigh University. This book, his fifth, grew out of a graduate course he taught at Lehigh. His other books include Algebra: An Approach via Module Theory (with W. A. Adkins).

Actions of Linearly Reductive Groups on Affine Pi Algebras (Memoirs of the American Mathematical Society)

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Actions of Linearly Reductive Groups on Affine Pi Algebras (Memoirs of the American Mathematical Society)
Nilolaus Vonessen
Manufacturer: Amer Mathematical Society
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Binding: Paperback

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Arithmetic of Finite Fields: First International Workshop, WAIFI 2007, Madrid, Spain, June 21-22, 2007, Proceedings (Lecture Notes in Computer Science)

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Arithmetic of Finite Fields: First International Workshop, WAIFI 2007, Madrid, Spain, June 21-22, 2007, Proceedings (Lecture Notes in Computer Science)

Manufacturer: Springer
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Binding: Paperback

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ASIN: 3540730737

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This book constitutes the refereed proceedings of the First International Workshop on the Arithmetic of Finite Fields, WAIFI 2007, held in Madrid, Spain in June 2007.

The 27 revised full papers presented were carefully reviewed and selected from 94 submissions. The papers are organized in topical sections on structures in finite fields, efficient implementation and architectures, efficient finite field arithmetic, classification and construction of mappings over finite fields, curve algebra, cryptography, codes, and discrete structures.

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Aspects of Galois Theory

Manufacturer: Cambridge University Press
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Binding: Paperback

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Galois theory is a central part of algebra, dealing with symmetries between solutions of algebraic equations in one variable. This collection of papers brings together articles from some of the world's leading experts in this field. Topics center around the Inverse Galois Problem, comprising the full range of methods and approaches in this area, making this an invaluable resource for all those whose research involves Galois theory.

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