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Billy Woods - mathematics
United Kingdom
Registrace 25. 02. 2020
[ANT15] p-adic integers: a primer, and an application (part 1)
The p-adic integers are pretty easy to define, but it's far from obvious what the point of them is, or how we should even think about them. In this video, I describe them as a practical tool: a collection of number systems that are related to the usual integers Z, but where solving equations can be much easier. It would be nice if we could port these results back over from Z_p to Z, but that's a topic for another video...
zhlédnutí: 5 952
Video
![[ANT14] Eisenstein polynomials and Strassmann's theorem, via a cubic example](http://i.ytimg.com/vi/R3eUz_Sm_vs/mqdefault.jpg)
![[ANT14] Eisenstein polynomials and Strassmann's theorem, via a cubic example](/img/tr.png)
[ANT14] Eisenstein polynomials and Strassmann's theorem, via a cubic example
zhlédnutí 884Před 2 lety
So far we've focused mostly on quadratic extensions, and have missed some interesting (although rather tricky and technical!) results as a consequence. Let's see a few of them.
![[ANT13] Dedekind domains, integral closure, discriminants... and some other loose ends](http://i.ytimg.com/vi/xESwB0nb7bU/mqdefault.jpg)
[ANT13] Dedekind domains, integral closure, discriminants... and some other loose ends
zhlédnutí 1,5KPřed 2 lety
In this video, we see an example of how badly this theory can fail in a non-Dedekind domain, and so - regrettably - we finally break our vow of not learning what a Dedekind domain is.
![[ANT12] Quadratic reciprocity and prime factorisation](http://i.ytimg.com/vi/5D7B0KIs3JA/mqdefault.jpg)
[ANT12] Quadratic reciprocity and prime factorisation
zhlédnutí 960Před 2 lety
In this video, we finally put all the pieces together and see how quadratic reciprocity can help us to factorise primes in quadratic extensions of Z. ANT books: Marcus, "Number Fields"; Ireland and Rosen, "A Classical Introduction to Modern Number Theory"... Keith Conrad's notes: kconrad.math.uconn.edu/blurbs/gradnumthy/quadraticgrad.pdf
![[ANT11] Quadratic Gauss sums](http://i.ytimg.com/vi/fBkIFjkROLU/mqdefault.jpg)
[ANT11] Quadratic Gauss sums
zhlédnutí 1,5KPřed 3 lety
Last video, we used the fact that √2 = ζ ζ⁻¹, for ζ an 8th root of unity, to tell us about the decomposition of rational primes in Z[√2]. In this video, we see that it is also possible to write √q as a sum of roots of unity for all *odd* primes q: in fact, we write down an explicit element τ whose square is ±q.
![[ANT10] Prime decomposition in Z[√2]](http://i.ytimg.com/vi/m-9Fjgp8BH0/mqdefault.jpg)
[ANT10] Prime decomposition in Z[√2]
zhlédnutí 1KPřed 3 lety
We return to our study of primes. We don't yet have a general method to find the primes of anything other than Z[i], and eventually, we'll want to know that; so let's start for now with the (relatively easy) example of Z[√2]. How can we tell when a (rational) prime splits, remains inert or ramifies after extension to Z[√2]? What difficulties present themselves? How far will these methods genera...
![[ANT09b] The Diophantus chord method](http://i.ytimg.com/vi/C4biRErh2Ow/mqdefault.jpg)
[ANT09b] The Diophantus chord method
zhlédnutí 965Před 3 lety
Cremona's tables: johncremona.github.io/ecdata/ Table for example 2: www.lmfdb.org/EllipticCurve/Q/576/c/3 Tables for example 3: www.lmfdb.org/EllipticCurve/Q/27/a/3 and www.lmfdb.org/EllipticCurve/Q/432/e/4 Table for the exercise: www.lmfdb.org/EllipticCurve/Q/1728/n/3
![[ANT09] Stacking oranges](http://i.ytimg.com/vi/zgJQD4c1wa4/mqdefault.jpg)
[ANT09] Stacking oranges
zhlédnutí 839Před 3 lety
In this video, we see how Pell's equation can be used to solve higher-order equations over the integers. We see another example of the importance of understanding which numbers are squares modulo each prime. The question was originally posed by Lucas: Nouvelles Annales de Mathématiques, série 2, tome 15 (1876), p144, question 1194: www.numdam.org/article/NAM_1876_2_15 144_1.pdf (in French) Meyl...
![[ANT08b] Square triangular numbers](/img/n.gif)
[ANT08] Continued fractions, Pell's equation, and units of Z[√d]
zhlédnutí 5KPřed 3 lety
A brief detour into an unexpected kind of "approximation by rationals" algorithm allows us to easily solve Pell's equation (often by hand!), and hence calculate the units in Z[√d] for positive integers d.
[ANT07] Units and logarithm space (+ bonus theorems)
zhlédnutí 1,3KPřed 3 lety
There are infinitely many units in Z[√2]. How can we write them down? How can we figure out their multiplicative structure? (Plus, now that we've come this far, a few theorems to make the link between this course and more traditional courses.)
[ANT06] Real and imaginary embeddings
zhlédnutí 1,5KPřed 3 lety
When we try to draw a real quadratic extension of Z in the complex plane, it collapses onto the real line - we don't get a lattice any more. We're going to prise it apart by drawing it on the real line in two different ways at once. We'll be able to recover a genuine notion of geometry, and we'll see how we can apply Minkowski's theorem to learn about arithmetic in these rings.
[ANT05] Minkowski's geometry of numbers
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Unsurprisingly, many of the pictures we've drawn are honest geometric objects, leaving them open to geometric attacks.
[ANT04] Counting ideal classes
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We don't always get unique factorisation. In those cases, can we measure how badly this uniqueness fails?
[ANT03] Ideal factorisation: recovering uniqueness
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[ANT03] Ideal factorisation: recovering uniqueness
[ANT01] Algebraic number theory: an introduction, via Fermat's last theorem
zhlédnutí 11KPřed 3 lety
[ANT01] Algebraic number theory: an introduction, via Fermat's last theorem
When part 2 ? 🙏
3:42
Is there any book/lecture note that is close to your approach? Please, inform, I need a written form.
There is, I think, a perfect sequel to this series: Dino Lorenzini, An Invitation to Arithmetic Geometry. It's not much more difficult but fleshes out the logic (such as what's in this video: why Dedekind domains? Discriminants? Ideal class groups?) REALLY thoroughly; has copious examples like this video series; and it gets all the way up to proving the Riemann-Roch theorem! I do think it's best to also own Hartshorne's Algebraic Geometry at the same time though: just the first few pages make what's in Lorenzini's book feel like they have a solid context, and might make parts of it make more sense.
@@SBWielinga Thank you very much for your informative comment.
The rational integers satisfy monic polynomials with integer coefficients of degree 1, e.g. x - 3 = 0, so the generalization to monic polynomials of higher degree for higher algebraic integers makes sense (if these linear polynomials were not monic, they're satisfied by non-integer, rationals numbers).
I Just came across this channel and it is fantastic! But I am a little worried that it may have stopped as this video was uploaded a year ago. Are you still going @billy_maths? I really hope that you are.
I looked up Frohlich and Tayler and it looks interesting. do you recommend any particular books on algebraic number theory?
Amazing! ❤
excellent video. I think I read a section about this (20 years ago) in Hardy and Wright but it didn't sink in.
this reminds me of pictures people use of the upper half plane of C (as a model for hyperbolic geometry).
I really appreciate the concrete approach and pacing. You explain enough that I can understand (maybe after pausing, rewinding in spots) but overall it is very clear and interesting.
I really enjoy the videos. I watched part 15 (on p-adic arithmetic), part 1 and part 2. These are well thought out, and challenging (I have had to stop, rewind, ...) but if I try I can learn ANT. It would be good to see part 16 (local to global I imagine). p.s. I am a mathematician who hasn't studied ANT.
Brilliant!
This video should be recommended to anyone who watched Stand-Up Matt's and lacked the context.
2.9K views and only 108 likes ??? It's a disgrace. It should be 2.9K likes....people don't know how to appreciate real quality work.
Excelente lecture. Many things contribute to a first class lecture: the graphics, the tempo of the presentation, the explanations, the tone of voice and a love of the subject. You have all these qualities. I just stumbled upin you lecture as I wanted to learn some about p-adic numbers. I'm sure glad I found your site. Please keep up the great work.
Dear Billy Woods, this lecture is phenomenal. I am so glad to have discovered your channel. I hope there to see much more of you in the future.
thank you so much for this serie of videos , tne content is really really enriching that's the least that I can say. I love how I see things totally differently now , known concepts make much more sense now , if you can do a video about Q_p and the projective line it would be marvelous. Again thanks a lot!!!
thank you so mcuh for this video series , this was really what I needed
Wait a minute Is 2s complement just a p adic representation?
Yep, it's a (truncated) 2-adic representation!
I also just learnt about this connect recently. Since 2s complement representation is a bit more familiar, I wish p-adic introduction materials start with this and expand to other primes.
@@billy_maths And because of how it is truncated, it only can hold integers.
Absolutely amazing video
15:26 "In an ideal world..." I see what you did there! 😂
I'm glad someone appreciates me! 😮💨
Very well presented, thanks!
Yeah, a buddy of mine told me about them.Tell you what, I'm never visiting Germany or Europe.
Excellent 👌 Can you make a video of the construction of p-adic integers Z_p algebraic way ? For example, we want to solve the equation $x^2=a$ (mod p) and then continue to divide by p and find successive solutions to get the p-adic integers.
How do you show p1p2 = (2), p3p4 = (3), ...?
I appreciate your videos so far. I am not finished yet. I have tried to read books on algebraic number theory, but could never really appreciate them because they do not provide good examples or motivation. Your videos do this.
I am used to seeing the Legendre symbol with parentheses, not square brackets.
This whole playlist was most illuminating, thank you
impeccable presentation
I love the video! I have a question, so my friend figured that any odd squared number minus one divided by 8 equals a Triangular number. I wanted to know if there is any equation could lead us to this pattern without just guessing randomly.
I tried to prove it but I'm not very sure about my answer. And I couldn’t ask my math teacher. So i squared (2k+1) giving me an odd squared number(n), n=4k^2+4k+1. Subtracting 1 from both sides, we get n - 1 = 4k^2+4k. - Factoring out 4k from the we get n - 1 = 4k(k + 1). Since k and k + 1 are consecutive integers, one of them is even. Therefore, 4k(k + 1) is divisible by 8. Im not sure ab the last step^^^
@@IRanamans Yes, you are correct. In the last step, just rewrite it as (2k+1)^2-1=4k(k+1)/8=k(k+1)/2 and since k is an integer this is exactly the form of the k'th triangular number.
your videos are fantastic ! thank you
This is a rare unique explanation. Thank you!
This explanation is wonderful and you are genius
Hello sir. I need to prove the following result "Class number 1 implies and implied by PID". What are the prerequisites from Alg. Number Theory to prove this result?
True almost by definition
@@cafeinst Hi, thanks for your reply but I have already figured it out😊😊😊😊
In 12:50 when you write gcd(2x,z) this gcd should be computed in the Gaussian integer. So to be careful one need to mention first that for integers a,b , the gcd(a,b) as integer is also a gcd for a and b as Gaussian integers.
Just to be a little pedantic, doesn't in general ring, two elements are relatively prime if their gcd is unit?
The best explanations on this subject I've ever found!
I'm really glad that i find your channel. your video quality is very good keep it up!!
This video is wonderful. Thank you!
Can you make such amazing playlist on algebraic geometry too? I am having a hard time with Hartshorne
Thank you for the very kind comment! I don't think I'll get around to that any time soon, but to my mind, Ravi Vakil has already done a far better job than I ever could: czcams.com/channels/y3u23mZE4TyW88yr6JLx9A.html
I had the completely unfounded notion this was an old one, so there must be a follow-up. Uploaded 7h ago 😐
Oops! Follow-up very soon, I hope. ;)
I've been secretly wishing for you to do a video introduction series to p-adics. Very happy to see you actually doing it :) Keep the good work up 👍
Thank you! More p-adic videos to come soon, I hope :)
Thank you for sharing! I have a question, on 20:45 why can you assume that unit to be 1?
Good question - I definitely skimmed over this detail! I seem to remember it's a tricky calculation involving modular arithmetic mod 9 (or equivalently mod π⁴), but I don't remember exactly how it goes. I'll have a think and see if I can get back to you!
Okay, I think I've got it. I'll write γ' = uc³ for some unknown unit u in Z[ω]. (There are only 6 units in Z[ω], so we have u = ±1, ±ω or ±ω².) Remember that α and γ are both divisible by π, and notice that y = (γ - α)/π, which we can rewrite as γ' - α/π. Now, since α is divisible by lots of factors of 3, so is α/π, and so we get y = γ' (mod 3), so that y = uc³ (mod 3). (This "mod 3" is actually modular arithmetic *inside Z[ω]*, i.e. arithmetic in the quotient ring Z[ω]/3Z[ω]. I wasn't assuming viewers would be familiar with this, which is probably why I omitted this detail!) Now, y is an integer and (by an assumption we made a few minutes earlier) is not divisible by 3, so either y = 1 (mod 3) or y = -1 (mod 3). Putting these together, we get ±1 = uc³ (mod 3). Write c = d + eω for some integers d and e: then c³ = d³ + e³ (mod 3), and again, as d³ + e³ is an integer, it must be 0, 1 or -1 (mod 3). Plugging this into the above equation, we get either ±1 = 0 (mod 3) (a contradiction) or ±1 = ±u (mod 3), which we can write slightly more neatly as 1 = u² (mod 3). Now we can just check the three cases directly: if u = ±ω, then 1 = ω² (mod 3), so ω²π = 0 mod 3, so π = 0 (mod 3), i.e. π is divisible by 3. But this can't be true: N(π) = 3 and N(3) = 9. Similarly u can't be ±ω². So it must have been ±1. (And if it's -1, then rewrite γ' = -1*c³ as 1*(-c)³.)
I love your style !! very great diagrams !
Excellent playlist! What software and tablet setup do you use to make these?
Thank you! In this video, I think I used my Microsoft Surface + pen, OneNote for handwriting, OBS to record video, Audacity to record audio, and OpenShot to edit. In my later videos, I'm mostly using Cavalry for the 2D animation and Lightworks for the editing. :)
amazing