I think I generally understood this section. The most difficult part was keeping straight in my mind what ring I am in. When I'm going from R to R[x] then to R[x]/(p(x)) then to doing congruence classes arithmetic in K. I just get a little jumbled up when reading examples so it would be nice to do a couple of more in class. I guess I also need a little clarification. Are we essentially saying that every polynomial must have a root therefore we will create larger rings specifically so that they contain this root when the polynomial was irreducible in the original field.
I was pretty excited to learn how the construction of the complex numbers works. It has always been difficult for me to understand what they are and were they came from. In fact the other day I was tutoring my little sister who is in her first semester of high school algebra. She was covering the types of number systems and I was having a difficult time telling her that the real numbers weren't really all the numbers because I couldn't explain where the complex numbers came from. She was confused and I was confused because I didn't have the background knowledge to comfortably tell her the reason. This goes back to the exact thing we were talking about in class where it is important for a teacher to have a deeper understanding of the topic than the student.
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