Chapter 12 of 24 - Fermat's Last Theorem and Unique Factorisation

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".

Chapter 11 of 24 - Rings and Polynomials

New mathematical objects! Introduction to rings. Introduction to fields. Introduction to polynomial rings over fields. An interesting return to long division (!) with polynomial division, and some great examples of shortcuts for factorising polynomials and finding roots. Can't help but feel we should have been taught about these shortcuts earlier. Not only do they seem really valuable, they're also surprisingly interesting and can probably be taught without rings and fields as a prerequisite.