MTH 233: Modern Algebra

Modern algebra rests on a foundation of set theory and logic. It is axiomatic in approach and investigates abstract structures independent of context. For example, in linear algebra the scalars are real numbers, the coordinate vectors and matrices have real entries. But, what properties of real numbers are actually used? Only the fact that we can combine real numbers by the operations of addition and multiplication, and that these operations satisfy certain axioms. Such an abstract structure is called a field. We don't need to confine ourselves to so-called real vector spaces. We can allow the field of scalars to be the field of rational numbers (so important in number theory), the field of complex numbers, the field of rational functions (complex analysis and differential equations), or even a finite field (coding theory). All the theorems of linear algebra, and thus all their applications, still hold!

The course will develop the abstract structures that evolved from attempts to solve concrete problems. One central concept this course is the notion of a group. Much of modern mathematics is based on this abstract structure. A group is a set on which is defined a single operation: a rule for combining elements of the set. Historically, the elements of the group were transformations and the group operation was composition. For example, the set of permutations of $n$ objects and the set of rotations of the plane are groups. In such disparate branches of mathematics and science as crystallography, quantum mechanics, relativity theory, graph theory and geometry, groups emerge wherever symmetry is present. The nature of the group tells us the degree of symmetry. The brilliant idea credited to Evariste Galois (killed in a duel in 1830 at the age of 20) of using symmetry groups to decide the solvability of equations captured the imagination of 19th century geometers, and was recently a pivotal idea in the proof by Andrew Wiles of Fermat's Last Theorem.

The course may also touch on rings and fields. Rings and fields are sets in which there are two operations (often called plus and times) To create a ring that is not a field, just abandon the axiom that says that every nonzero number has a multiplicative inverse, i.e., throw out division. The simplest ring is the ring of integers (1 divided by 2 is not an integer), but examples of interesting rings range from the finite rings of modular (clock) arithmetic to the ring of matrices and the ring of linear differential operators studied at Smith in the physical chemistry course.