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
Algebraic Number Theory notes by Professor Robert Ash: faculty.math.illinois.edu/~r-.... These notes are quite thorough and they have a lot of core material needed for algebraic number theory.
Algebraic Number Theory videos by Billy Woods: • [ANT] An unorthodox in... . These videos are very well made and they have all the core intuitions in them. I personally loved how visual they are; they make the concepts feel a lot more visceral.
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).
-----
WORKS CITED
The data of class numbers for the cyclotomic number rings was from here:
oeis.org/A055513
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:
www.lmfdb.org/NumberField/
-----
MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
www.vincentrubinetti.com/
THANK YOUs:
Extra special thanks to Davide Radaelli and Vivek Verma (@vcubingx) for helpful conversations while making this video. you guys rock!!
Follow me!
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)

[@ethanbove629]

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

[@jurjenbos228]

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.

[@ophello]

I mean…typically he doesn’t really explain that much. It’s a bunch of notes but no real deep explanations. This video is nice, but the other ones are disappointingly vague.

[@mrtaurho8846]

@@ophello Well, isn't that kinda the point of this channel, to give some general ideas of what is out there? Especially as other channes usually focus on more basic topics. You can't really expect a full-blown lecture in ~20 minutes. Sure, vagueness can be problematic but for the time frame given I think they do a great job of conveying ideas and motivations. Plus, I think the last part on resources is a great addition if you're interested in the actual math behind their videos.

[@SubAnima]

​@@ophello I think the point of Aleph 0 is to make content that pure math majors (like myself) would only expect to cover at a fourth year level or even higher, extremely interesting. Like I'm thinking of taking Algebraic Number Theory next semester and have a rough notion of what it's about but watching this video just makes it tangible what the field is about, why it's beautiful etc. If I wanted detailed lectures, sure I'd just take the course at my uni, or read the books Aleph 0 was suggesting. But when I'm on KZbin I really just want to be inspired and go "wow that is so so cool" without getting bogged down with proofs which would make this video an hour long. That's what class/self-study is for. And on the inspiration front, I think this channel couldn't have done better. I know now why Algebraic Number Theory is interesting and have some motivation to study it in the future.

[@mrtaurho8846]

@@SubAnima Well said!

[@angelmendez-rivera351]

@@ophello Actually, I think it is the other way around. This video has many conceptual inaccuracies and mistakes, but his other videos so far are all excellent.

[@mrtaurho8846]

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.

[@Aleph0]

hey thanks! that's really interesting - introducing Iwasawa theory via regular primes could be a nice perspective. I'm guessing you start by talking about Q(zeta_37), and then talk about the class number of Q(zeta_37^n) for all n? There's an interesting story there. Also thanks for the reference to Milne's notes: I'll add them to the description :)

[@mrtaurho8846]

@@Aleph0 I found it always a curious fact that Lamé's historical non-proof works just fine for regular primes. It then quite naturally leads to the concept of class numbers. From there one can get to the cyclotomic tower, to the general study of class numbers and ultimately to the main setup of Iwasawa Theory. I admit that's it's not a perfect road but gives an interesting perspective nonetheless. However, it usually requires some more mathematical background than your video :D Milne's notes do also include many, many examples and a computational angle through Pari/GP. A recommendation of my ANT professor. It was quite refreshing using it besides good ole Lang... :) EDIT: I just realized my initial comment seemed to suggest that going through FLT towards Iwasawa Theory is the superior approach; quite the contrary is what I meant.

[@theflaggeddragon9472]

@@mrtaurho8846 The road from Lame, to Kummer, to Iwasawa theory and modern algebraic number theory is one of my favorite developments in all mathematics. I've ended up studying Galois representations and p-adic Hodge theory motivated precisely by that story!

[@MarcusAndersonsBlog]

@@Aleph0 @19:27 That Washington's book starts with FLT is hardly coincidental. After Wiles, FLT is an essentially pointless observation that is a restatement of (what I refer to as) the primitive number matrix {ℵ1}[] (aka Pascals triangle). {ℵ1}[] was already known in prehistory (BC), and independently discovered in different cultures, and available to Fermat who lived before and after Pascal. {ℵ1}[] defines all calculation, and renders the first 360+ pages of Principia Mathematica somewhat "unnecessary" (as per Ramsey). A 1991 proof (by E.Post) of a 1951 paper by A.Moessener was published on KZbin's Mathologer channel last year. This (unwittingly) shows a simple solution to FLT in plain sight based on the definition 1+1=2. Post didnt notice it and neither did Mathologer. Post posted (pun) his proof 1 year before Wiles, so strictly speaking Post was 1st past the post (pun#2) with his accidental and very simple marvelous proof of FLT which was intended to prove "Moessener's miracle" of slicing up {ℵ1}[] into powers of N by showing that each power was derived from the one before it. Thus FLT is shown to be true in the course of Post's 1991 proof using a simple non-Wiles method.

[@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!

[@Aleph0]

thanks gabe! Hope you found the video helpful :)

[@locusf2]

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!

[@joda7697]

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!

[@theflaggeddragon9472]

Fantastic video as usual and I just want to say how much I appreciate you making advanced math videos for free on KZbin. 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

[@wolfgangfrech5942]

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.

[@levav8]

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

[@hvok99]

This was phenomenal, would love to see this channel grow. While the last few years has really been some captivating math content creators come to KZbin, the depth and clarity of your videos is really appreciated. I am currently working on through a calculus 2 course and am particularly interested in the math of infinite series, I find the geometric interpretation of these series really wonderful and have begun to see the power of representing objects in the complex plane. There is so much in your videos that is beyond me but it is definitely content like yours that motivate my independent study. Happy to support on Patreon 🎉

[@odysseus231]

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.

[@rbnn]

This is extraordinarily well-presented and interesting. Making these abstract notions at this level concrete and enticing is amazing.

[@Octa9on]

Wonderful! I already knew a smidgeon about ideals, but until now I'd never seen a clear explanation of where they come from and why they are important. Thanks!

[@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

[@yamiyugi2894]

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

[@edawgroe]

Great video! Glad I found this channel. "This book is a graduate textbook, so it is terse" -> first chapter is "Fermat's last theorem". That killed me hahaha

[@lovaaaa2451]

I am absolutely astonished by the beauty of these videos. Thank you for your work, much love !

[@hasanathasan4651]

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

[@saraanderson2538]

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

[@agustinlawtaro]

Great! Thank you. I really like that you give numerical examples and references.

[@JM-us3fr]

Wow, excellent summary of algebraic number theory. Glad to have learned something new!

[@juliangst]

Very interesting video. Definitely on the same level as 3blue1brown, Mathologer or the other big math channels!

[@telnobynoyator_6183]

I think it's more similar to Michael Penn actually

[@punditgi]

Truly majestic explanations. Keep, 'em comin'! 👍

[@CallMeIshmael999]

This was a very ambitious idea for a video! I'm glad you do what you do.

[@wilderuhl3450]

Such an amazing video as always. Keep up the good work

[@3of7tricom34]

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

[@Aleph0]

hey thanks! hope you found the video helpful :)

[@stighemmer]

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

[@yoyokojo651]

Amazing! I love the effort put into these videos!

[@kapoioBCS]

Amazing work! Looking forward to your next video!

[@pandoraedwin1222]

This is incredible and mindblowing. Thank you so much.

[@mrl9418]

Congratulations on the video and on the 100k !

[@kevinportillo9882]

Congrats on the 100k! Keep it up, 3b1b has some serious competition here

[@MaximQuantum]

Woah, it’s so fascinating.

[@keryannmassin5596]

Such a nice video, as always!

[@lindsay3917]

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

[@Aleph0]

thanks Lindsay! I hope you found the video helpful :)

[@itolstukhin]

typo on 7:34. one ideal should be generated by 2 and 1 - sqrt(5). great video btw!

[@pseudolullus]

Extremely good

[@johnlinley2702]

The most fun I’ve had in years, and this in a dark time.

[@manueldelrio7147]

Thank you! Your channel is definitely one of my top three!!

[@Aleph0]

thanks manuel!

[@erictao8396]

This channel is amazing!

[@beauthetford7608]

Yay great video! I love everything about your videos. They always reignite my excitement for abstract algebra. You even influenced how I write the Greek letter Zeta, something I have always messily done. I like to call that third property of an ideal, the "sticky property" since everything from the ring gets "stuck" inside, if you combine with an ideal element. Every element in the ideal has a layer of fly paper that catches ring elements!

[@beauthetford7608]

Also I'm not sure I ever learned about the importance of class number before. So if my ring is a principal ideal domain, does that mean it has class number 0?

[@endian1516]

great video I watched the whole thing!

[@Mrpallekuling]

I read about ideals in a book by Harold M Edwards and it was a challange. I should have seen this video much earlier. Great presentation. Thanks.

[@jogloran]

Looking forward to supporting you on Patreon! Thanks for the amazing maths content.

[@Aleph0]

thanks daniel! I appreciate it :)

[@Drachensslay]

Remarkable. Awesome video

[@ichigo169_g7]

This math is definitely above my level. So sorry if this is obvious but what would the class number of just the normal integers be?? You don't need to do anything extra so I would think 0 but my gut is saying 1

[@ericvosselmans5657]

at 4:58 you say "...closed under addition and multiplication..." while writing "I is closed under + and -" So I am guessing that that should have been ".. closed under addition and subtraction.."

[@aniketeuler6443]

Finally ur back 🤩🤩

[@mastershooter64]

will you make a video on functional analysis?

[@jakobovergaard6302]

Fantastic content!

[@Icenri]

Thanks for your work!

[@cafebrasileiro]

Wow, great video! Made me go through my Algebraic Structures notebook from 2 years ago haha. I think I finally got an intuitive idea of Ideals... Ideally though :P

[@KStarGamer_]

I'd be super interested in a video on the Kummer-Vandiver conjecture.

[@dylanparker130]

This was really interesting & congratulations on your success!

[@Aleph0]

thanks for watching, dylan!

[@superman39756]

Your videos are awesome

[@TheConair92]

I am a graduate student at UMD and Prof Washington is great! Have talked to him many times

[@Aleph0]

lucky you! I am more than a bit jealous :)

[@user-ss6nb4no5o]

Id like to take time to say thank you big man i know nothing about mathematics and i feel am in a good place since am thinking of going back to school and doing applied mathematics

[@vlyoni]

Fantastic video!

[@JojiThomas7431]

Didn't get it quite at first. Will have to watch again.

[@TheOneMaddin]

I would have guessed that the initial reason to introduce ideals has been to define quotient rings. They are the equivalent to normal subgroups, or linear subspaces.

[@mrtaurho8846]

In a todays course on Abstract Algebra, yes. Historically, no. The ideal numbers of Kummer (which eventually became ideals as we know them) were introduced in response to the failure of unique factorization as described in the video. The historic circumstances are quite interesting actually. Lamé proposed a proof of Fermat's Last Theorem based on the wrong assumption of unique factorization in certain rings (the cyclotomic rings). Kummer observed that in these rings unique factorization can fail badly which led him to consider ideal numbers instead. Dedekind then formulated the notion of an ideal based on Kummer's ideal numbers.

[@motiveautomorphic3987]

No. The concept of rings were introduced much later than ideals.

[@TheOneMaddin]

@@mrtaurho8846 definitely interesting. Thanks

[@cycklist]

What a superb video 👍

[@tejarex]

In general, if int n = a*a + b*b, it can be factored in Z[i] as (a + bi)*(a - bi) = (-a + bi)*(-a - bi) = (b + ai)*(b - ai) = (-b + ai)*(-b - ai). So Z primes 2 = 1*1 + 1*1, 5 = 2*2 + 1*1, 13 = 3*3+2*2, 17 = 4*4+1*1, etc, can be non-trivially factored in 4 ways (2 for 2).

[@franciscogallardo5558]

I love your videos!

[@feynmanmafia2615]

Very cool video!

[@zathrasyes1287]

Awesome good presentation!

[@manuelrodriguez5293]

Great video! Thanks

[@Spacexioms]

I can't believe I understood this and the possibilities just seem astronomical now

[@RohitSingh-nm9wd]

Upload morr video please. It helps me to push people to do math by letting them watch your videos

[@alexeifando747]

Much appreciate you!

[@nocturnhabeo]

WOO A NEW VIDEO

[@MrRyanroberson1]

8:30 there's a sign error. When first introducing ideals, you say 'addition and multiplication' where you mean addition and subtraction. There are a few small errors like this

[@angelmendez-rivera351]

There are some major errors too, conceptual ones at that too.

[@timelsen2236]

8:30 isn't an error. The ideal ring is closed under both, just as for real ideals. How this works for cyclotomic rings , meaning roots of unity , has a problem with addition. Only for the 3rd roots does the 2 complex roots add to -1, so mod(2) give the real root 1 back. Cos(120)=-.5 here ,is irrational for all other Nth roots. Say for the 6th roots, the product of conjugates, 1st and 5th roots is sqrt(3) by 30,60,90 basic trig. Trig is dominated by irrational results, other than .5, so for the 6th roots to make a ring the radius must be sqrt(3), which then gives 3 for the product of the 1st and 5th roots so introducing (3) into the ring. Do any commenting here, have more than a superficial understanding of these issues? Try the 7th roots and please let me know the modulus for this to be closed under addition of the discrete vector space!

[@Pika250]

The gist was that primality depends on the ring, where an element p may be prime to R but not to S, where R is a subring of S (say p = ab with a and b in S \ R), and that prime ideals are a generalization of prime elements that guarantees unique factorization. Yes, it is true that not every unique factorization domain is a principal ideal domain, and such a counterexample would be beyond the scope of this, but this makes the proof that every principal ideal domain has unique factorization easy.

[@Taylor-rx4yb]

I wish I saw this video when I first encountered algebraic number theory (or even algebra for that matter)!

[@shalvagang951]

can you also make videos on stuffs like clifford algebra nd geometric algebra weyl algebra i really like your explanation very nice

[@luyombojonathan7715]

Great stuff

[@ericvosselmans5657]

Some very good content, extremely well explained. As a curious math layman i wonder : Are there more ways to factor 5 if you consider more dimensions? I.e. are there perhaps 4 ways to factor 5 in Quaternion space?

[@angelmendez-rivera351]

The notion of factorization is not meaningful in the quaternions, since the quaternions are not a commutative ring. Factorization is only a meaningful concept in integral domains. An integral domain is a commutative ring where the only zero divisor is 0 itself.

[@Moinsdeuxcat]

Awesome!

[@Juniper-111]

at 9:49, the ideal on the left be (2, 1-√(-5))

[@biblebot3947]

Can you make a video on the symmetric product?

[@berry4862]

Super interesting. Could you explain or reference, how to multiply expressions of for example the form (a, b+c sqrt(-5)) etc. in general? I mean like (2)*(3)=(6) but for the non-principal forms. I could guess, but the proper definition would be interesting.

[@dstahlke]

I think you would just multiply each number from the first with each number from the second. Then for aesthetic purposes, remove redundant terms from the result (e.g. terms that are products or linear combinations of other terms).

[@angelmendez-rivera351]

@@dstahlke No, you want to keep the linear combinations. If I and J are ideals, then their product I·J is an ideal {a(0)·b(0) + a(1)·b(1) + ••• : a(i) in I and b(j) in J}.

[@dstahlke]

@@angelmendez-rivera351 Yes, the ideal contains all the elements {ab | a in I, b in J} and all linear combinations thereof. But it can be specified using a minimal set of generators. Since an ideal is closed under addition and scalar multiplication, the ideal generated by a, b, c (for example) will automatically contain all linear combinations of those elements. So when listing a minimal set of generators, you'd want to not write any that are linearly dependent on the others.

[@timelsen2236]

In reply to the many not realizing they didn't understand , 8:30 isn't an error. The ideal ring is closed under both addition and multiplication, just as for real ideals. How this works for cyclotomic rings , meaning roots of unity , has a problem with addition. Only for the 3rd roots does the 2 complex roots add to -1, so mod(2) give the real root 1 back. Cos(120)=-.5 here ,is irrational for all other Nth roots. Say for the 6th roots, the product of conjugates, 1st and 5th roots is sqrt(3) by 30,60,90 basic trig. Trig is dominated by irrational results, other than .5, so for the 6th roots to make a ring the radius must be sqrt(3), which then gives 3 for the product of the 1st and 5th roots so introducing (3) into the ring, as well as (sqrt 3), for the 0th root! Do any commenting here, have more than a superficial understanding of these issues? Try the 7th roots and please let me know the modulus for this to be closed under addition of the discrete vector space!

[@diribigal]

I wanted to join the Patreon but it seems I can't be a patron and give money below the amount of the "early videos" tier (at least on mobile?)

[@mathamateur7033]

Thanks a billion times.... 🙏🙏🙏

[@billy_maths]

Great video!

[@Aleph0]

right back at you! I loved your ANT videos :)

[@a5110123]

I have a little question at 9:49 Do you mean that "A pair of ideals is said to be in the same class if they differ by a scalar multiple which is in the field of fraction of the ring" ?

[@koushikhalder6390]

Thank you very much.

[@patrickhanlon932]

I'm impressed that youtube thinks that I'm smart enough to understand any of this!

[@anonimmors1925]

when video about some complex manifolds or bundles?

[@rudihelderton5150]

Great video! Thanks! It's not clear to me how the ideal 6 factors into those 4 different prime ideals. Can someone explain this?

[@angelmendez-rivera351]

The video explained how to multiply them together.

[@fyadmohammed8856]

A question.. By taking the multiples of 2 multiplied by the multiples of 3 , did you mean the linear combination of both sets??

[@Jordan-zk2wd]

Iwasawa's approach reminds me a bit of quantum mechanics/quantum field theory, in terms of the idea of summing over all histories as comparing to summing over all these ideals. I have have heard of the techniques developed in QM being applied to knot theory in a similar way (looking at an infinite family of knots or all possible knots simultaneously), so I dunno if maybe there was any direct influence in this case as well, but a cool parallel in the philosophy/approach nonetheless.

at 4:58 you said closed under addition and multiplication but had closed under addition and subtraction on the screen

[@NonTwinBrothers]

In other news: Aleph 100,000 just broke 0 subcribers!! Way to go

[@superman39756]

Keep going

[@zy9662]

Is there a relationship between the class number and the Galois group? For example, the extensions with non abelian Galois group have certain kind of class numbers?

[@user-sk4kg4hr3k]

7:37, I believe there is a mistake, there are two identical rings (2, 1+sqrt-5),i think one of them should be with minus sign

[@mihirsheth6]

Fantastic video! Just a small correction: the power of p in the class number of Z adjoined with p-roots of unity = \lambda*n + u, not p^{\lambda*n + u}.

[@Aleph0]

thanks for the correction, Mihir. I've added it to the description :)

[@NoNTr1v1aL]

2:01 Don't worry Flash. Your secret's safe with me.

[@DDranks]

Is 7:36 supposed be (6) = (2, 1+sqrt(-5))(2, 1-sqrt(-5))(3, 1+sqrt(-5))(3, 1-sqrt(-5)) I think you have one minus sign flipped!

[@DDranks]

Also, as a bit of critique: the consept of ideals was freshly introduced at this point. It would be very helpful, and a service to your audience, to visualize each of the ideals that goes into factorialization of (6). It's also stated a bit too implicitly what does a multiplication of two non-principal ideals even mean. (A cartesian product?) Also, as multiplication up to complex numbers is commutitative, I'd expect a discussion about whether just switching two objects like "ideals" is allowed...