I felt like I was understanding most things about irreducibility until halfway through section 4.5. And then they brought in prime numbers as criteria for reducibility and I got completely lost. I read the rest of the section and did not understand hardly any of the proofs. Since I am a visual learner it will be really helpful for me to see examples of how to use the criteria mentioned in the section to figure out if a polynomial is irreducible. Unfortunately, it got worse as I read section 4.6. Beginning with Lemma 4.28 and the new notation they introduce I essentially didn't understand any of of the section. I really cannot understand the notation and it doesn't help that I am unfortunately relatively unfamiliar with complex numbers.
Because I was so confused, I found it difficult to make a lot of connections between this section and other things that I have learned. I do think it's great though that there are more ways to determine if a polynomial is irreducible because the methods that we were using before are not very efficient in many cases. It is also interesting that there have never been any formulas derived to find the roots of polynomials above fifth degree. My inner desire to solve unsolvable problems makes me want to study this, but knowing that many geniuses have examined this gives me little hope that I would get anywhere.
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