1. The hardest part of this section is understanding how to use the principles and theorems and apply them to actually proving if something is a field. I was starting the homework for this section and a lot of the problems ask you to prove is something is a field. I did not know how to apply what I had just read with the theorems to be able to compute a proof to show is something is a field. So this is the most difficult part of the reading.
2. I think it is really cool that Theorem 5.11 helps to explain problems and doubts that people had back in history with complex numbers and negative numbers. It is hard for me to connect Abstract Algebra with normal, useful math. I don't feel like it has a place in mathematics. However, this section showed me how Abstract Algebra is used to explain simple math concepts like negative and complex numbers -- things that would not be as common had there not been Abstract Algebra and this theorem to help people grasp the concepts.
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