Algebraic number theory - an illustrated guide | Is 5 a prime number?

This video is an introduction to Algebraic Number Theory, and a subfield of it called Iwasawa Theory. It describes how prime numbers factor in infinite towers of number rings.

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Minor corrections:

at 4:58: I should have said "closed under addition and *subtraction*" instead of "closed under addition and multiplication." The text on the screen is correct.
at 16:32: Instead of the "power of p dividing the class number", this should read the "p-part of the class number".

SOURCES and REFERENCES for Further Reading!

This video is a quick-and-dirty introduction to Algebraic Number Theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.

(a) ALGEBRAIC NUMBER THEORY

PREREQUISITES: The prerequisites for learning Algebraic Number Theory are: group theory, ring theory, and Galois theory. It's possible to get a basic non-rigorous feel for the subject without these prerequisites, which is what I tried to do in this video. But if you want to know the details (for example: you might have asked: what *exactly* is a number ring?), then these prerequisites are essential. To learn these prereqs, check out the previous video on this channel, "How to self study math", where there are a bunch of resources to learn these prereqs.

(b) IWASAWA THEORY

Introduction to Cyclotomic Fields by Lawrence Washington: This book is AMAZING! To see Iwasawa theory in action, skip directly to chapter 13, Iwasawa's theory of Zp extensions. (You don't need to read the book in sequential order because the chapters are largely independent.) The proof of this theorem is just miraculous.

PREREQUISITES: Algebraic Number Theory (that is, the previous section).

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WORKS CITED

The data of class numbers for the cyclotomic number rings was from here:
This list is only for cyclotomic number rings (Z adjoin a p-th root of unity) where p is a *prime* number.

The two examples of class numbers (class numbers 100 and 2000) was from the L-functions and Modular Forms database:

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MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.

THANK YOUs:

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Twitter: @00aleph00
Instagram: @00aleph00

Intro: (0:00)
Number Rings: (1:41)
Ideals: (4:46)
Unique Factorization: (8:55)
Class Numbers: (11:41)
Iwasawa Theory: (14:53)
Thank you!: (18:37)
Learning Resources: (18:49)
Patreon: (19:45)

[ethanbove]

Honestly one of my favorite parts about taking high-level math classes is how these videos are slowly making more and more sense. I love the satisfaction of having just learned about prime ideals and then seeing it here

[jurjenbos]

As a Ph.D. in mathematics, I must say I now better understand ideals. Kudos!

[SubAnima]

The king is back! Such a cool video, your distillation of high level maths into such a clear format inspired me to make a channel doing the same for biology. Can’t wait for the next one.

[GabeWeymouth]

As an educator not familiar with this material, I have to say that your introduction was really great! Simple. Clear. Surprising. It immediately set the hook!

[locusf]

I saw in the documentary of Fermat's Last Theorem that Wiles used Iwasawa Theory somehow but I never knew what it was until now. Thank you for widening my perspective about mathematics!

[joda]

This isn't actually that confusing, despite your insistencies that it is.
The visual explanation paired with the algebra that you did really made the concept crystal clear.
Although i get that for you who had no such video, it must have been hard and/or confusing to learn.

You make the subject of pure math more easily accessible and you should be proud of yourself for that!
Thank you for these amazing videos!

[theflaggeddragon]

Fantastic video as usual and I just want to say how much I appreciate you making advanced math videos for free on YouTube. This is a truly neglected space I've been thinking of throwing my hat into. Keep them coming!!

In algebraic number theory no less! My favourite topic

[wolfgangfrech]

35 years ago, I gave up on algebra lectures at "ideals of a ring". Today, you rekindled my curiosity with a (kind of) hands-on problem of unique prime factors.

[mrtaurho]

I never thought it possible to introduce the basic idea of Iwasawa Theory with essentially nothing to build on. But you did it anyway!
When attempting to explain it to someone I usually tried to going throughout FLT and its solution for regular primes (Washingtons first chapter) as motivation. This might be a better approach :D

P.S. I think J.S. Milne's notes on ANT are an excellent addition to your list.

[stighemmer]

In twenty minutes you taught me something I spent days without getting in school. Thank you!

[levav]

Such a fun video! I just finished a course on Algebraic number theory last semester - now I can finally tell people outside the math world what I was doing XD

[ShaolinMonkster]

Very good presentation. I lost you at class number.
I know this topic is hard so im impressed i got up to class number and understood everything

[saraanderson]

I love the way that this video captures the beauty and artistic side of math, in a way that standard lectures don't. I wish all math was presented this way! math is a journey, not a destination

[odysseus]

The fact that you give clear sources and encourage further, more rigorous reading and learning is really what sets you apart. You are one of the best maths channels for budding maths students. Thank you so much.

[hasanathasan]

i've been watching since the 10k days, and youre already at already at 100K! thats incredible!

[yamiyugi]

Glad you are still making videos, keep up the great work

[lindsay]

This was great! I'm in analytic number theory and this made the ideal language make so much more sense. Thanks!!

[beardymonger]

Thank you, thank you, thank you!!!
EDIT: This approach, example based with historical background, is the one only one I can learn from. One of my favourite math writers who uses this way is Edwards, for example Galois Theory.

[oftricom]

Absolutely fantastic! Really well explained. And great of you to give additional resources for others to learn this as well!

[sebastiannrregaard]

What a great video on algebraic number theory! What a way of explaining the class group and number without really defining it! I love it <3