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Algebra with Galois theory. Notes by Albert A. Blank. Republishing of the book published 1947 under the title Modern higher algebra. Galois theory. (English) Zbl 1131.12001

Courant Lecture Notes in Mathematics 15. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 978-0-8218-4129-7/pbk). vi, 126 p. (2007).
Emil Artin was one of the great mathematicians of the twentieth century. His influence on the development of mathematics in the first half of the last century was tremendous, both by means of his many outstanding contributions to research in various areas and by the high level and excellence of his teaching and expository writing. It is fair to say that Emil Artin was not only one of the founders of modern abstract algebra and algebraic number theory, but also one of the pioneers in modern mathematical teaching and writing. His famous books and lecture notes on different topics, captivating by their unrivaled conceptual clarity, aesthetics, and inspiring power likewise, were widely used for several decades and virtually set the standards for their numerous successors in our times. As Richard Brauer once put it, each of Emil Artin’s expository writings presented a novel approach to the respective subject, with always new ideas and results, and it was a compulsion for him to show the beauty of the subject in its purest form to the reader.
Among Emil Artin’s most popular short books were his lecture notes “Modern Higher Algebra. Galois Theory”, written down by Albert A. Blank and first published in 1947 by the Courant Institute of Mathematical Sciences of New York University. Now, because of both its historical significance and its continual popularity among instructors and students for several decades, the Courant. Institute came to the decision to republish this volume under the new, nowadays more proportionate title “Algebra with Galois Theory”, thereupon making it available for new generations according to its deserts.
The booklet under review is precisely this republished version of Emil Artin’s Courant Lecture Notes from 1947, without any alterations concerning the original text, but in modern printing.
In seven chapters, assuming no prerequisites whatsoever and starting from scratch, Artin quickly and masterly develops those fundamental concepts of abstract algebra that are necessary to both formulate and treat the basics of modern elementary Galois theory.
Chapter 1 briefly introduces groups, whereas Chapter 2 just as briefly explains rings, fields, and vector spaces. Chapter 3 provides the fundamentals of polynomial rings over a field in one variable, including their ideal theory and factorization properties. Chapter 4 turns to polynomial equations of degree \(n\) and their related theory of field extensions. Chapter 5 forms the main body of the book and discusses, on about 30 pages, the essentials of modern elementary Galois theory up to the fundamental theorem, following Artin’s own elegant approach via group characters as in his other classic “Galois Theory” (Zbl 1053.12501). This chapter also includes a brief introduction to the theory of finite fields. Chapter 6 deals with integer polynomials, Eisenstein’s irreducibility criterion, and primitive roots of unity. Chapter 7 focusses then on applications of Galois theory to both the theory of algebraic equations and the ancient problems of geometric constructions by ruler and compass. The solution of equations by radicals, Steinitz’s Theorem on simple field extensions, and Abel’s Theorem are the central topics of this final chapter.
It is still amazing to see how elegantly, masterly and comprehensibly Emil Artin taught such a topic like Galois theory sixty years ago. Using the modern framework of abstract algebra, presenting each argument in its simplest and purest form, and stripping the theory from any unnecessary ballast, he manages to teach the subject in an utmost accessible, inspiring and profound manner. Many contemporary textbooks on Galois theory need hundreds of pages to reach a comparable depth of exposition, alas with much less lucidity, elegance, and pedagogical effect. This is why one can still learn from a grandmaster like Emil Artin, and why the present classic will maintain its everlasting significance also in the future.

MSC:

12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
12F10 Separable extensions, Galois theory
01A75 Collected or selected works; reprintings or translations of classics
12E12 Equations in general fields
12E20 Finite fields (field-theoretic aspects)
12F05 Algebraic field extensions

Citations:

Zbl 1053.12501
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