Abstract algebra

Abstract algebra is one of the core classes in the undergraduate mathematics curriculum. The most important objects studied in abstract algebra are

  • Groups
  • Rings
  • Fields

While typically defined abstractly, in terms of axioms, groups are collections of symmetries of objects, and so we begin with a study of symmetry. For example, given a cube in space, what rigid motions (reflections, translations, rotations, and their compositions) can we do that take the cube onto itself? There are 48 such rigid motions in total, and 24 if we disallow reflections. The 24 without reflections form a very important group known as S_4, and all 48 form a more exotic group, called a hyperoctahedral group. In all cases, symmetries of objects form groups. Eventually, we will turn to a more axiomatic study of groups. The theory of groups is very beautiful in its own right, but it surprisingly turns out to be a good setting for parts of number theory, and we will see several results in number theory pop out of group theory. For example, Fermat’s Little Theorem and Euler’s generalization involving the totient function are special cases of a simple theorem in group theory. Some combinatorial problems can also be solved using group theory; for example, we can count the number of Rubik’s cube configurations, which would be far more difficult without group theory.

After that, we will study rings, which are algebraic structures that allow addition, subtraction, and multiplication, but not division in general. The integers form a ring, and in a certain sense the prototypical ring. Ring theory is the natural setting for primes: we can define primes in any ring, and, rather surprisingly, understanding the structure of other rings can give us valuable insight into the study of primes over the integers. One possible application we might do is to prove Fermat’s Last Theorem for n<37. (n=37 is much more difficult than any smaller case, and we will be able to understand why.) Another possibility is to look at algebraic curves, which may be thought of as pictorial representations of certain rings.

Finally, we study fields, which are generalizations of the rational or real numbers: we can add, subtract, multiply, or divide, as long as we do not divide by 0. With only a small amount of field theory, we will be able to prove the impossibility of two of the classical Greek problems: duplicating a cube and trisecting an angle. If time permits, we will also look at Galois theory, the tool needed to show that quintic (and higher-degree) equations cannot, in general, be solved in terms of radicals. Galois theory is widely considered to be one of the most spectacular branches of mathematics.