Serge Lang’s 20 Best Math Books for Math Students

Serge Lang's 20 Best Math Books for Math Students | Abakcus

Serge Lang was a mathematician and activist who spent the majority of his career teaching at Yale University. He was of French and American descent, and his contributions to number theory and mathematics textbooks, particularly the popular Algebra, have brought him a great deal of notoriety.

In 1960, he was honored with the Frank Nelson Cole Prize and participated in the Nicolas Bourbaki group. As a member of the activist community, Lang led campaigns against the Vietnam War and the nomination of political scientist Samuel P. Huntington to a position in the National Academies of Science. Both campaigns were ultimately successful.

Lang was a prolific writer of mathematical texts, and he would frequently finish one during his time off over the summer. Most of them are for graduate students, and he also authored calculus textbooks and helped Bourbaki produce a book on group cohomology.

The Mathematical Contributions of Serge Lang

Lang’s Algebra work was a highly influential graduate-level introduction to abstract algebra that went through a number of revised editions throughout its publication history. According to the citation for his Steele Prize, “The graduate-level study of algebra was revolutionized by Lang’s Algebra… It affects all math textbooks written after it, including graduate students.”

During my mathematics studies, I could use many of Serge Lang’s books, and they were an incredible asset to the team. Because of this, I have compiled a list of the top 20 math books recommended by Serge Lang for students studying mathematics in college.

If you are a college math student, you should also check 40+ Interesting Math Books from Springer Undergraduate Series.

Math!: Encounters with High School Students contains a collection of well-known mathematician Serge Lang’s dialogues with students. With his lively style, Serge Lang treats the students as his own and shows them something of the essence of mathematical thinking. The encounters between Lang and the students were taped and, therefore, authentic and lively. The book introduces a fresh and novel approach to teaching, learning, and enjoying mathematics. The book is of great interest to teachers and schools.

You might be taken aback if someone were to tell you that mathematics can have a very beautiful side. But you should be aware that there are those people who spend their entire life doing mathematics and who develop mathematics in the same way that a composer makes music. In most cases, when a mathematician finds a solution to a problem, this results in the creation of many more problems that are fresh and stunning in the same way that the original problem was. Of course, a lot of the time, these issues are pretty challenging, and, just like in other fields of study, they can only be comprehended by individuals who have researched the topic extensively and have a solid grasp of the material.

In 1981, Jean Brette, who was in charge of the Mathematics Section at the Palais de la Decouverte (formerly known as the Paris Science Museum), invited me to present at the Palais. I had never before presented such a conference to an audience that was not familiar with mathematics. This presented me with a challenge: was it possible for me to explain to such a large audience on a Saturday afternoon what it meant to practice mathematics and why one does mathematics?

When I refer to mathematics, I mean purely mathematical concepts. This does not imply that pure mathematics is superior to other forms of mathematics; however, since a few other people and I are involved in pure mathematics, I am now worried about those individuals. Math has a poor reputation, which may be traced back to the earliest levels of education. The term is put to use in a wide variety of settings. To begin, I was required to provide a brief explanation of these potential contexts, as well as the one with which I intended to engage.

Math Talks for Undergraduates

Serge Lang has been giving presentations to first-year mathematics students at the university for many years. Math Talks for Undergraduates focuses on particular topics in mathematics that may be broken down to a level understandable by students who have already taken calculus. Lang is currently presenting a collection of those conversations as a book, written in a conversational tone. Members may deliver the lectures of the teaching staff. However, it would be even more beneficial if they were delivered by students participating in seminars that were organized and managed by the students themselves.

Mathematics students at the high school or college level will greatly benefit from reading Basic Mathematics by Serge Lang. It gives students a solid grounding in the fundamental concepts of mathematics. It prepares them to move on to more difficult areas of study, such as calculus, linear algebra, and other related subjects. The information is provided understandably, and the author builds concepts in such a way as to demonstrate how one topic can relate to and develop into another.

The book Fundamentals of Differential Geometry introduces fundamental ideas in differential topology, differential geometry, and differential equations, as well as some of the most important fundamental theorems in all three fields. Although this book is a continuation of the author’s earlier work, Differential, and Riemannian Manifolds, the emphasis has shifted away from the general theory of manifolds and toward general differential geometry.

Together with Linear Algebra, Undergraduate Algebra is part of the algebra program’s curriculum geared toward students in their first year of college. The separation of linear algebra from the other basic algebraic structures conforms to all current undergraduate education trends, and I agree with these trends. I have ensured that this work can stand on its own, rationally speaking.

However, it is highly recommended that students first get familiar with linear algebra before moving on to the more abstract concepts of groups, rings, and fields, as well as the methodical development of the fundamental abstract features of these concepts. Because I attempted to make this book stand alone, there is some duplication between it and the book Linear Algebra. I begin by defining vector spaces, matrices, and linear maps, then proceed to demonstrate the fundamental properties of each one.

The present book has the potential to be utilized for either a single semester or an entire academic year’s worth of instruction, with the inclusion of Linear Algebra as an option. I believe it is vital to do the field theory and the Galois theory; more importantly, I would say that to do much more group theory than we have done here. Specifically, I think doing the field theory and the Galois theory is important. A section on finite fields demonstrates characteristics that come from general field theory and characteristics that are unique because of characteristic p. These interdisciplinary disciplines have recently become significant in coding theory.

Author Serge Lang defines algebraic geometry as the study of systems of algebraic equations in several variables and the structure that one can give to the solutions of such equations. The study can be carried out in four ways: analytical, topological, algebraic, geometric, and arithmetic.

This volume offers a rapid, concise, and self-contained introductory approach to the algebraic aspects of the third method, the algebraic geometric. The treatment assumes only familiarity with elementary algebra up to the level of Galois theory.

Starting with an opening chapter on the general theory of places, the author advances to examinations of algebraic varieties, the fundamental theory of varieties, products, projections, and correspondences. Subsequent chapters explore normal varieties, divisors and linear systems, differential forms, the theory of simple points, and algebraic groups, concluding with a focus on the Riemann-Roch theorem. All the theorems of a general nature related to the foundations of the theory of algebraic groups are featured.

Undergraduate Analysis, written by Serge Lang, is an introduction to Analysis that is logically self-contained and is appropriate for students who have already completed two years of calculus. The features of uniform convergence and uniform limits in the setting of differentiation and integration are at the heart of this book. These concepts are discussed in detail throughout the text.

Real and Functional Analysis is intended to serve as a text for an analytical course taken by graduate students in their first year of study. It tackles the same issues as elementary calculus but does so in a way appropriate for individuals employing it in subsequent mathematical explorations. In this sense, it covers the same topics as elementary calculus. Because the organization strives to avoid extensive lines of logical connection, its chapters are largely autonomous from one another. This makes it possible for a course to leave out content from some chapters without affecting how content from succeeding chapters is presented.

All articles loaded
No more articles to load
Ali Kaya


Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.