Finally at the half-way mark. This chapter took a long time to get through and the assignment was especially challenging. Again, I think I'll have a lot to write about once that is marked and returned.
The sections of chapter 12 were Fermat's Last Theorem and Diophantine equations, Integral Domains, Euclidean Domains, and Unique Factorisation Domains. There are subtle differences between all these different types of domains, even though the description of each one might very well just look like a construction of a polynomial.
For example, in integral domains, a prime and an irreducible are different things, but in Euclidean domains and UFDs, they're the same thing.
Also, one thing to note that I realised isn't in my handbook... norms can be constructed for polynomial domains. The norm is just the polynomial's degree. But this can only be done if the domain is a field.
Right, wow. What a journey. On to the next book. "Metric Spaces 1".