Extracurricular Mathematics II: Electric Boogaloo

John von Neumann, a major contributor in the field of mathematics, once told one of his students, “…in mathematics you don’t understand things. You just get used to them.”  This is sometimes how I’ve felt during my first two years of advanced education in Mathematics.  I also think I have  gained an understanding of some advanced topics in Real Analysis, Abstract Algebra and Algebraic Number Theory, and Measure Theory, thus increasing by a little bit my understanding of Mathematics.  But certainly I have had to get used to the above topics and mathematical concepts before I could fully understand and appreciate them.

I’ve managed to get through four sections (out of 12 or 13) of my summer book, \underline{\textrm{Primes of the Form}\,x^2 + ny^2}.  I’ve gotten slowly used to the concepts presented in these first four sections: quadratic forms, the form class group, and genus theory.  I also have come to some understanding of these concepts.  But I’ve also hit a wall in Section Five.

The main topic introduced in Section Five is class field theory.  However, in the presentation of this topic, the reader is assumed to be familiar with – or used to – Galois Theory.  Apparently, Galois Theory was developed to find solutions of polynomials of degree three or higher.  I have enough background to understand the basic definitions of Galois Theory, but I will need some time to get used to Galois Theory.  My summer book also came due at the library, so I think mine was a good stopping point.  Thus my summer project has given birth to a side project.

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