The most difficult part of this material for me was conceptually understanding the extension field and trying to visualize it. I am also confused when they say the F[x]/(p(x)) contains a root of p(x), if that root is a class or just a number. And if it's a class, is it the same type of class as the classes in F. Also, I was wondering if every elements in F is a congruence class, or not necessarily.
The most interesting part of this material for me was trying to think about why it would be useful to have the congruence-class ring. I also thought it was interesting to see how many levels the properties of integral domain, irreducibility, etc. can be carried out and still apply. It seems like you can always just keep finding bigger and bigger areas to work in, which kind of boggles my mind. I wonder if these applications have anything to do with different dimensions in space.
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