I had a little bit of a hard time understanding the idea of the extension field K. The proof of 5.12 wasn't too hard to understand, but I had to break it down a little bit farther and think about how if p(x) is an irreducible factor of f(x) and p(x) has roots, then any roots it has are also roots of f(x).
Reflective:
Well I feel like this is what I've been saying this whole chapter, but this is a lot like the stuff we did back in chapter 2. What's cool though is that while F[x]/p(x) is a field just like how Zp is a field, F[x]/(p(x)) also contains a root of p(x), which is obviously not possible in Zp.
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