Day 8 (11/06/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:04 - 12:03)

            Today I continued to study from the textbook that Professor Dolbin had given me. I finished a brief chapter on an algebraic notion called “fields,” which are groups of elements under two operations. An example is the set of real numbers under addition and multiplication. After studying this branch of algebra, I moved on to studying Galois Theory, which studies the relationships between fields, field extensions, and groups. This was used to eventually show that there are NO ways to solve general equations of degree 5 or higher; for instance, the famous Quadratic Formula can solve any quadratic equation, but Galois showed that there are no such formulas for polynomials of degree 5 or higher. I find it fascinating how this is one of the greatest examples of how mathematics had proven the fact that something does not exist, which really sounds impossible at first thought. This will likely be included in my research later on. Professor Dolbin frequently checked up on me and guided me when needed, and I have already learned more than I possibly could have by myself in such a short amount of time. The past couple of weeks had been invaluable to me, as they showed me that I really can survive while doing just math all day, which is what most mathematicians in the field do. I am grateful that Professor Dolbin is willing to spend so much time to mentor me in this manner.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Next, in Professor Farag’s class, he taught us more about ring theory, which is another structure in algebra like “groups” or “fields,” as well as how rings relate to those other structures. Throughout his lecture today he emphasized the need to stop ourselves from invoking our previous knowledge on “numbers,” and the need to stick with the axioms given only. This mindset is crucial for the study of abstract algebra, as I learned through the various classes that our intuition may not always hold. For instance, x plus y might not always equal y plus x in a lot of algebraic structures. Boy, does this class give me a headache!