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Fields and Galois Theory de John M Howie

Fields and Galois Theory

Autor John M Howie
en Limba Engleză Paperback – 19 dec 2005

Ediția de față, apărută inițial în 2005 și revizuită în 2007, propune o introducere concisă în studiul corpurilor algebrice, acele seturi fundamentale în care operațiile raționale pot fi executate fără restricții. Observăm că John M Howie a ales o abordare deliberat non-istorică, preferând să construiască riguros fundamentele algebrei abstracte moderne pornind de la concepte contemporane, spre deosebire de modul în care Galois sau Abel au dezvoltat inițial aceste idei prin permutări.

Considerăm că structura volumului este optimă pentru un curs universitar de licență, progresând natural de la inele și corpuri către domenii de integritate și polinoame. Ne-a atras atenția modul în care autorul utilizează teoria Galois pentru a clarifica probleme geometrice clasice, precum „cuadratura cercului”, oferind în același timp un fundament solid pentru înțelegerea grupurilor și a ecuațiilor. Lucrarea completează perspectiva oferită de Galois Theory de Ian Stewart, adăugând o rigoare axată pe structura abstractă și eliminând detaliile istorice extinse care definesc textul lui Stewart. Totodată, spre deosebire de A Field Guide to Algebra de Antoine Chambert-Loir, volumul de față oferă o cale mai directă către aplicațiile geometrice și include o secțiune utilă de soluții pentru exerciții.

În contextul operei sale, Fields and Galois Theory păstrează stilul clar și aplicat pe care John M Howie l-a demonstrat în Pratially Ordered Groups. Dacă în lucrarea anterioară autorul punea accent pe tehnicile speciale ale disciplinelor de nișă, aici reușește să sintetizeze subiecte complexe de algebră pură într-un format accesibil, sub 250 de pagini, fără a sacrifica profunzimea demonstrațiilor matematice.

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Specificații

ISBN-13: 9781852339869
ISBN-10: 1852339861
Pagini: 226
Ilustrații: X, 226 p. 22 illus.
Dimensiuni: 179 x 238 x 18 mm
Greutate: 0.4 kg
Ediția:2005. Corr. 2nd Printing 2007 edition
Editura: SPRINGER LONDON
Locul publicării:London, United Kingdom

Public țintă

Lower undergraduate

De ce să citești această carte

Această carte este recomandată studenților la matematică ce doresc o introducere directă și clară în teoria Galois. Cititorul câștigă o înțelegere profundă a modului în care grupurile abstracte pot rezolva probleme vechi de secole legate de polinoame și geometrie. Este un instrument excelent pentru studiul individual, datorită structurii logice și a soluțiilor incluse, fiind ideală pentru pregătirea examenelor de algebră superioară.

Descriere scurtă

Fieldsaresetsinwhichallfouroftherationaloperations,memorablydescribed by the mathematician Lewis Carroll as “perdition, distraction, ugli?cation and derision”, can be carried out. They are assuredly the most natural of algebraic objects, since most of mathematics takes place in one ?eld or another, usually the rational ?eld Q, or the real ?eld R, or the complex ?eld C. This book sets out to exhibit the ways in which a systematic study of ?elds, while interesting in its own right, also throws light on several aspects of classical mathematics, notably on ancient geometrical problems such as “squaring the circle”, and on the solution of polynomial equations. The treatment is unashamedly unhistorical. When Galois and Abel dem- strated that a solution by radicals of a quintic equation is not possible, they dealt with permutations of roots. From sets of permutations closed under c- position came the idea of a permutation group, and only later the idea of an abstract group. In solving a long-standing problem of classical algebra, they laid the foundations of modern abstract algebra.

Cuprins

Rings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.

Recenzii

From the reviews:
“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.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)
"The author wrote this book to provide the reader with a treatment of classical Galois theory. … The book is well written. It contains many examples and over 100 exercises with solutions in the back of the book. Sprinkled throughout the book are interesting commentaries and historical comments. The book is suitable as a textbook for upper level undergraduate or beginning graduate students." (John N. Mordeson, Zentralblatt MATH, Vol. 1103 (5), 2007)
"To write such a book on a widely known but genuinely non-trivial topic is a challenge. … J. M. Howie did exactly what it takes. And he did it with such vigour and skill that the outcome is indeed absorbing and astounding. … Every paragraph has been scheduled with utmost care and the proofs are crystal clear. … the reader will never feel forlorn amidst brilliant theorems, which makes the book such a good read." (J. Lang, Internationale Mathematische Nachrichten, Issue 206, 2007)
"Howie’s book ... provides a rigorous and thorough introduction to Galois theory. ... this book would be an excellent choice for anyone with at least some backgound in abstract algebra who seeks an introduction to the study of Galois theory. Summing Up: Highly recommended. Upper-division undergraduates; graduate students." (D. S. Larson, CHOICE, Vol. 43 (10), June, 2006)
"The latest addition to Springer’s Undergraduate Mathematics Series is John Howie’s Fields and Galois Theory. … Howie is a fine writer, and the book is very self-contained. … Iknow that many of my students would appreciate Howie’s approach much more as it is not as overwhelming. This book also has a large number of good exercises, all of which have solutions in the back of the book. All in all, Howie has done a fine job writing a book on field theory … ." (Darren Glass, MathDL, February, 2006)
"The book can serve as a useful introduction to the theory of fields and their extensions. The relevant background material on groups and rings is covered. The text is interspersed with many worked examples, as well as more than 100 exercises, for which solutions are provided at the end." (Chandan Singh Dalawat, Mathematical Reviews, Issue 2006 g)

Textul de pe ultima copertă

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.

Caracteristici

Modern and student-friendly introduction to this popular subject: it takes a more "natural" approach and develops the theory at a gentle pace with an emphasis on clear explanations Features plenty of worked examples and exercises, complete with full solutions, to encourage independent study Includes supplementary material: sn.pub/extras