Info from ETS:
The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level.
Approximately 50 percent of the questions involve calculus and its applications — subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors.
About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.
The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.
CALCULUS — 50%
Material learned in the usual sequence of elementary calculus courses — differential and integral calculus of one and of several variables — includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations and other branches of mathematics.
ALGEBRA — 25%
Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics
Linear algebra: matrix algebra, systems of linear equations, vector spaces, linear transformations, characteristic polynomials and eigenvalues and eigenvectors
Abstract algebra and number theory: elementary topics from group theory, theory of rings and modules, field theory and number theory
ADDITIONAL TOPICS — 25%
Introductory real analysis: sequences and series of numbers and functions, continuity, differentiability and integrability, and elementary topology of R and R^n
Discrete mathematics: logic, set theory, combinatorics, graph theory and algorithms
Other topics: general topology, geometry, complex variables, probability and statistics, and numerical analysis
The above descriptions of topics covered in the test should not be considered exhaustive; it is necessary to understand many other related concepts. Prospective test takers should be aware that questions requiring no more than a good precalculus background may be quite challenging; such questions can be among the most difficult questions on the test. In general, the questions are intended not only to test recall of information but also to assess test takers' understanding of fundamental concepts and the ability to apply those concepts in various situations.
Our take on the test:
At GP Group, we know that if you want to get into a graduate mathematics program, you need the highest score possible on the GRE Subject Math Exam. Entry into graduate level math programs is extremely competitive, and the most important factor for admission is an applicant’s score on the GRE Subject Math Exam. A good score sets you apart from the crowd and secures the competitive advantage that you need to take the next step toward realizing your academic goals and pursuing your professional dreams.
The test, which is created and administered by ETS, is not only rigorous, but each test taker is graded in relation to all of the others. This style of grading, known as curve grading, means that you cannot simply rely on a baseline level of knowledge to score well on the test, like most of the exams you took in school. In order to score well, you must be better than the other test takers. In this way, it is more like a race than a conventional test. If the other test takers are extensively preparing, you must do the same.
But the subject matter tested on the exam is extremely specialized, and good test prep materials are scarce. Moreover, even if you find some off the shelf test prep materials, they will not be able to address your individual weak points and deficiencies, and they certainly will not be able to help you with test-taking strategies designed to maximize your individual potential.
But SubjectMath.com can! We can show you how. We have helped students everywhere to be better prepared.
It will not be easy, but neither is graduate school. So when you decide that you just will not be satisfied with a score that is just “alright,” or “good enough,” It is time for you to reach out to us. When you decide you are ready for the challenge, We will be here for you, ready and waiting to help you “get the perfect score!”
Which textbooks should I use?
Remember that succeeding in this test is the ability to solve a particular type of questions fast. Reviewing textbooks may be time-consuming. I would recommend viewing all the video modules, concentrate on the results rather than the proofs and solve a lot of practice exams. Whenever you encounter a topic that you have never studied, then you can turn to textbooks. In such case, these are our recommendations:
Calculus - Apostol
Linear Algebra - Hoffman and Kunze
Abstract Algebra - Dummit & Foote
Topology - Munkres Complex
Analysis - Ahlfors
For the additional topics (ODE, Real Analysis, Probability, Set Theory, Etc..) you need just a few basic results so diving into textbooks would not be a good idea in our experience.
The book by Princeton Review also does a good job in reviewing the basic topics for the exam.