A concrete introduction to higher algebra

This text is intended for a one-or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.

American Mathematical Monthly, 1999

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The College Mathematics Journal, 2004

Let Q and R be the fields of rational and real numbers respectively. Recall that a real number r is algebraic over the rationals if there is a polynomial p with coefficients in Q that has r as a root, i.e., that has p(r) = 0. Any college freshman can understand that idea, but things get more challenging when one asks about arithmetic with algebraic numbers. For example, being the roots of x 2 − 3 and x 2 − 20 respectively, the real numbers r 1 = √ 3 and s 1 = 2 √ 5 are certainly algebraic over the rationals, but what about the numbers r 1 + s 1 , r 1 s 1 and r 1 s 1 ? As it happens, all three are algebraic over the rationals. For example, r 1 + s 1 is a root of x 4 − 46x 2 + 289. But how was that polynomial constructed, and what rationalcoefficient-polynomials have r 1 s 1 and r 1 s 1 as roots? Students who take a second modern algebra course will learn to use field extension theory to show that the required polynomials must exist. They will learn that whenever r and s = 0 are algebraic over Q, then the field Q(r, s) is an extension of Q of finite degree with the consequence that r + s, rs and r s are indeed algebraic over Q (see ). However, one would hope that students would encounter more elementary solutions for such basic arithmetic questions. Furthermore, one might want to know how to construct rational-coefficient polynomials that have r + s, rs and r s as roots and thereby obtain bounds on the minimum degrees of such polynomials.

2015

The tone and level of mathematical sophistication of these two chapters is considerably different in these two chapters from those in the others. Much more background is expected from the reader interested in these sections.

Preface 1 The present book is an English translation from my book with the same title in Arabic language which are based on my lectures given to students of various colleges studying mathematics. In designing this course, the author tried to select the most important mathematical facts and present them so that the reader could acquire the necessary mathematical conception and apply mathematics to other branches. Therefore, in most cases we did not give rigorous formal proofs of the theorem. The rigorousness of a proof often fails to be fruitful and therefore it is usually ignored in practical applications. The book can be of use to readers of various professions dealing with applications of mathematics in their current work. The subject matter is presented in a very systematic and logical manner. It contains material which you will find of great use, not only in the technical courses you have yet to take, but also in your profession after graduation, as long as you deal with the analytical aspects of your field. In designing this book the author tried to select the most important mathematical facts and present them so that the reader could acquire the necessary mathematical conception and apply mathematics to other branches. This book consists of seven chapters. Chapter 1, "Sets -Relations -Functions" in abstract algebra, Chapter 2 contains the "Groups" as one of the main subjects. In Chapter 3 , we will discuss "Permutation Group" as a practical part and very useful in Linear Algebra. Chapter 4 presents "Isomorphism" which a fundamental part, and has many applications. Chapter 5, contains "The Natural Numbers" and how to extend the natural numbers up to real filed. Chapter 6 contains " Rings -Fields" as the second main subjects to abstract algebra. Chapter 7, which deal with "Continuation on Groups".

Contemporary Mathematics, 2015

2010

Let F be a number field. There are many interesting things we can compute about F: Invariants: maximal order OF, class group Cl(F), units U(F), higher algebraic K-groups, Dedekind ζF... Subfields: Galois group, lattice of subfields. Extensions: build L/F, e.g given explicitly by primitive elements or implicitly via Kummer or class field theory. Invariants thereof (e.g in class field towers). Basic operations: elementary operations on elements and ideals of OF, mostly multiplications (at least in class field theory). XIV e Rencontres Arithmétiques de Caen (20/06/2003) – p. 2/19Setup (2/4) For most of these problems, there exist efficient algorithms, deterministic or randomized, possibly assuming some deep conjecture (GRH, density of friable elements in appropriate sets...), possibly giving a wrong result with small probability in an appropriate model, possibly not an algorithm at all but usually giving sensible results... But there are a number of pitfalls, especially when the degree...