Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography
Traditionally, mathematics has been separated into three main areas; algebra, analysis and geometry. Of course there is a great deal of overlap between these areas. For example, topology, which is geometric in nature, owes its origins and problems as much to analysis as to geometry. Further the basic techniques in studying topology are predominantly algebraic. In general, algebraic methods and symbolism pervade all of mathematics and it is essential for anyone learning any advanced mathematics to be familiar with the concepts and methods in abstract algebra. This is an introductory text on abstract algebra. It grew out of courses given to advanced undergraduates and beginning graduate students in the United States and to mathematics students and teachers in Germany.
We assume that the students are familiar with Calculus and with some linear algebra, primarily matrix algebra and the basic concepts of vector spaces, bases and dimensions. All other necessary material is introduced and explained in the book. We assume however that the students have some, but not a great deal, of mathematical sophistication. Our experience is that the material in this can be completed in a full years course. We presented the material sequentially so that polynomials and field extensions preceded an in depth look at group theory. We feel that a student who goes through the material in these notes will attain a solid background in abstract algebra and be able to move on to more advanced topics. The centerpiece of these notes is the development of Galois theory and its important applications, especially the insolvability of the quintic.
After introducing the basic algebraic structures, groups, rings and fields, we begin the theory of polynomials and polynomial equations over fields. We then develop the main ideas of field extensions and adjoining elements to fields. After this we present the necessary material from group theory needed to complete both the insolvability of the quintic and solvability by radicals in general. Hence the middle part of the book, Chapters 9 through 14 are concerned with group theory including permutation groups, solvable groups, abelian groups and group actions.
Chapter 14 is somewhat off to the side of the main theme of the book. Here we give a brief introduction to free groups, group presentations and combinatorial group theory. With the group theory material in hand we return to Galois theory and study general normal and separable extensions and the fundamental theorem of Galois theory. Using this we present several major applications of the theory including solvability by radicals and the insolvability of the quintic, the fundamental theorem of algebra, the construction of regular n-gons and the famous impossibilities; squaring the circling, doubling the cube and trisecting an angle. We finish in a slightly different direction giving an introduction to algebraic and group based cryptography.
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