This section got a little weird at the end. It was difficult for me to wrap my mind around the Euclidean Algorithm for polynomials. I can see how I would be able to do that very easily. In fact it seems unless you do the polynomial division twice for each step you won't be able to move to the next equation. Also I was confused by the proof of theorem 4.5. I tried to read it and I think that I can't wrap my head around what it looks like to write a gcd in the form f(x)u(x)+g(x)v(x) and so the proof didn't make sense.
I really enjoyed this section though because I had never thought about treating polynomials like integers with factors, gcds, and linear combinations. I think that it is more fun to do factors of polynomials. I also really thought that it was clever to redefine the gcd to be unique by introducing the idea of a monic polynomial. It should have seemed obvious to me that this would be the solution to infinitely many factors but it was a huge revelation when they told us how to fix this problem. I enjoy how much arithmetic is in the discussion of polynomials.
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