Integral Extensions (Commutative Algebra 14)

In this lecture, we'll introduce integral extensions. We'll prove some basic properties: in particular, how it interacts with chains of extensions, localizations and quotients. We'll also discuss some examples that arise in number theory and algebraic geometry.

[amrita]

I liked the examples that used a picture to show integral extensions like the parabola to line and the non-integral Hyperbola to line. It makes it easier to understand when there's a picture involved.

[luna]

Most of this was in Algebraic Number Theory, which is a great review, but the geometric picture is a helpful second point of view. (Also it's so helpful to have the timestamps added in the video!)

[hansrajjangir]

We have covered most of the content of this lecture in the Algebraic number theory course, which I am taking with Dr. Schmidmeier in this semester. It was very interesting to see those example of integral and non-integral from geometry. This lecture provided me a picture of how these concepts are related to each other.

[matthewtrang]

I am also taking algebraic number theory this semester. From what I have seen, integral extensions play a central role in the study of the rings of algebraic integers in number fields. It is sort of one of the conditions for those rings to be well-behaved (Dedekind domains). Also, I like the graphical examples in the middle of this videos. They gave me some intuitions about the connections of integral extensions and algebraic geometry.

[theresabuscemi]

The beginning of this lecture was a good review of Algebraic Number Theory, since we have covered the concept of integral extensions in detail in that course; even some of the theorems overlapped. However, this lecture expanded on my prior knowledge by introducing the geometric application of integral extensions, as well as results regarding preservation of integrality by quotients and localizations, which I had not previously seen.