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

[@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

[@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.

[@jurjenbos228]

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

[@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 :)

[@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.

[@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.

[@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!

[@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

[@nicholasserrambana]

Really nice comfortable pacing in this video

[@stighemmer]

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

[@yamiyugi2894]

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

[@HFVXML]

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

[@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

[@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 :)

[@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.

[@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!

[@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 :)

[@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 🎉

[@rbnn]

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

[@sebastiannrregaard5849]

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

[@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

[@andrer.6127]

Yo, this is dope af. I watched this video like 10 times and 6 was the perfect number to use as an example.

[@manueldelrio7147]

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

[@Aleph0]

thanks manuel!

[@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.

[@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

[@agustinlawtaro]

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

[@kevinportillo9882]

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

[@JM-us3fr]

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

[@punditgi]

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

[@lovaaaa2451]

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

[@jogloran]

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

[@Aleph0]

thanks daniel! I appreciate it :)

[@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 :)

[@wilderuhl3450]

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

[@itolstukhin]

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

[@johnlinley2702]

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

[@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

[@benjiusofficial]

Brooooooo, this is exactly the stuff I was missing from Ring Theory. The second you explained 2*3=6, my mind immediately went 'linear transform' and everything up the chain clicked.

[@yoyokojo651]

Amazing! I love the effort put into these videos!

[@mrl9418]

Congratulations on the video and on the 100k !

[@keryannmassin5596]

Such a nice video, as always!

[@Drachensslay]

Remarkable. Awesome video

[@endian1516]

great video I watched the whole thing!

[@pandoraedwin1222]

This is incredible and mindblowing. Thank you so much.

[@dylanparker130]

This was really interesting & congratulations on your success!

[@Aleph0]

thanks for watching, dylan!

[@aniketeuler6443]

Finally ur back 🤩🤩

[@CallMeIshmael999]

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

[@kapoioBCS]

Amazing work! Looking forward to your next video!

[@MaximQuantum]

Woah, it’s so fascinating.

[@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?

[@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.."

[@erictao8396]

This channel is amazing!

[@mastershooter64]

will you make a video on functional analysis?

[@pseudolullus]

Extremely good

[@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!

[@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!

[@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).

[@Juniper-111]

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

[@fgrideau]

isnt there a mistake at 11:05 ? (3,1-sqrt(82)) and (6,4+2sqrt(82)) are not proportional

[@KStarGamer_]

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

[@superman39756]

Your videos are awesome

[@NonTwinBrothers]

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

[@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...

[@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.

[@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

[@Galinaceo0]

Also, altho for rings ideals happen to coincide with the things you can quotient by, this is not true for more general algebraic structures. For example, in monoids, there is also a notion of ideal, but not every quotient is obtained from an ideal.

[@Langasotobe]

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

[@biblebot3947]

Can you make a video on the symmetric product?

[@GreatestPhysicistOfAllTime]

The pair of product ideals (2,1+√(-5))(3,1-√(-5)) and (2,1-√(-5))(3,1+√(-5)) don't seem to correspond to any prime number factorization of 6 in the ring Z[√(-5)]. May I understand that not all cross-multiplications of prime ideals give rise to prime number factorization?

[@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.

[@TheOneMaddin]

You said repeatedly "an ideal is closed under addition and MULTIPLICATION". Do you mean SUBTRACTION? Otherwise it's just a subring, isn't it?

[@Aleph0]

yes, thanks for the correction! I'll add it to the description.

[@patrickhanlon932]

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

[@jakobovergaard6302]

Fantastic content!

[@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

[@IKMbKal]

Hey! I think there is a typo at 7:36. One of the ideals (2, 1+sqrt(-5)) should be written as (2, 1-sqrt(-5)).

[@RohitSingh-nm9wd]

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

[@franciscogallardo5558]

I love your videos!

[@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.

[@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.

[@Spacexioms]

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

[@feynmanmafia2615]

Very cool video!

[@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.

[@mathamateur7033]

Thanks a billion times.... 🙏🙏🙏

[@АндрейВоинков-е9п]

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

[@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?)

[@billy_maths]

Great video!

[@Aleph0]

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

[@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

[@ClumpypooCP]

The class number of normal integers would be 1!

[@fyadmohammed8856]

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

[@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" ?

[@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 :)

[@Taylor-rx4yb]

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

[@Icenri]

Thanks for your work!

[@nocturnhabeo]

WOO A NEW VIDEO

[@theultimatereductionist7592]

8:26 Don't you mean 1 minus square root of -5 for the third ideal on the right? You have (2,1+sqrt(-5)) twice

[@budoo88]

Good video 👍

[@zathrasyes1287]

Awesome good presentation!

[@manuelrodriguez5293]

Great video! Thanks