Artisian
@Artisian@lemmy.world
- Comment on What is class field theory? 1 day ago:
Not a number theorist, but the wikipedia reads ok for me, so I’ll give an attempt. Answer based on the AMS’s Translated Math Monographs 240, by Kazuya Kato et. al…
A sample of the questions class field theory wants to address: a) Which primes p are the sum of 2 squares, p=a^2 + b^2? b) What about other formulae, say eg p=a^2 +2b^2? c) Consider a Galois extension. Take a prime ideal P in the smaller ring. For which primes does this ideal factor when we look at the larger ring? d) When is the factorization square free (unramified)? e) What’s the smallest cyclotomic extension that contains sqrt(M) for a given M?
If we look at the integers, you may already know the answers to several of these! And they all have something kinda magic in common. For (a), for example, the primes that are the sum of 2 squares are exactly those with p = 1 mod 4. For example, 5=2^2+1^2, yet 7 cannot be written as a sum of two squares. The answer to question (b) is similar! We can do it exactly when p=1,3 mod 8.
For ©, for concreteness let’s take the extension of the rationals Q to the rationals with a square root of -3, Q(sqrt(-3)). The prime ideal (7) factors as (7, 1-sqrt(-3)) (7, 1+sqrt(-3)) (a product of two distinct prime ideals; unramified), as do the ideals (13), (19), (31), and (37). But (5), (11), (17), (23) and (29) all don’t. Perhaps you notice a pattern: p=1 mod 3 ? factors. p=2 mod 3? doesn’t. There’s also a unique ramified prime, (3) = (sqrt(-3))^2. There will generally only be a finite number of ramified primes. Do a dozen more examples and you’ll notice a spooky pattern: the ramified primes seem to show up in the modulus (in this example, 3 was ramified and the factorization pattern works mod 3. If 7 and 23 are ramified, the factorization cases will work modulo 7*23=161). [Quadratic extensions are not special btw; the factorization of (p) in Q(zeta_5) (Q with a 5th root of 1) depends on p mod 5.]
On the face of it, why would modular arithmetic be the relevant condition? And why does the modulus seem to care about ramification?
A major result of Galois theory is that there’s a correspondence between subgroups of (Z/NZ)^* (integers modulo N under multiplication) and intermediate field extensions between Q and a cyclotomic extension Q(zeta_N). Prime ideal ramification and factoring can be stated in terms of this correspondence. Further, they show that every finite abelian extension of Q lives inside some Q(zeta_N). This result lets us explain all of (a)-(e). Generalizing it is one of the big motivations of class field theory. If we start not with Q, but with say Q(sqrt(-3)), what still holds? What is the right generalization of cyclotomic extensions and (Z/NZ)^*?
My understanding is that this program is quite successful. There’s a replacement for both that’s only somewhat more technical/tedious, and that gives similar results. One of the bigger successes is generalizing ‘reciprocity’ laws (the quadratic case is often taught in undergrad number theory; it’s about the surprising fact that p is a square mod q depends on if q is a square mod p).
- Comment on [Meta] Could we consider avoiding political topics in this community? 6 months ago:
As one of the few folks who have asked such questions, I obviously am against. I don’t think the dedicated pol communities are particularly good for honest questions about platforms/political figures; everything in those spaces feels like it’s being intentionally spun (even in discussions) in a way that this community does not. (Also, several of the communities you suggest as pol discussion places are… just not? Extremely few questions, most the posts are headlines, discussions don’t seem to happen much. Some feel closer to a curated feed of cringe.)
I do agree it could become an issue, and that would justify some division, perhaps tags? But I don’t think it is currently very unpleasant, and it will almost certainly get better in 2 months (at least short term).
- Comment on What does a federal ban on price gouging look like? 6 months ago:
I think the scary thing is if it takes the suppliers more than 3 days to figure that out. Companies oftentimes can last 3 days without food (and rarely fix things very quickly at any scale).
- Comment on What does a federal ban on price gouging look like? 6 months ago:
That one seems kinda scary - if inflation was 6% and something wasn’t sold at any profit, all stores would stop selling it. (This is true for most food.)
- Comment on What does a federal ban on price gouging look like? 7 months ago:
Agreed, that would be.
But the most they could have done is 308% instead of that 300%, and I think they managed to get lots and lots of small stores to do it at the same time.
- Comment on What does a federal ban on price gouging look like? 7 months ago:
I’ll note that grocers record profits are orders of magnitude less than the price increases. Maybe somebody is getting rich off of the price increases, but I’m pretty sure Walmart is not.
- Comment on What does a federal ban on price gouging look like? 7 months ago:
I’ll note that grocers seem to have made very little profit per American in the last few years; Walmart made ~$70 off each of us last year, which seems incompatible with the price increases I’ve been seeing…
- Submitted 7 months ago to [deleted] | 28 comments
- Comment on Why are people downvoting the MediaBiasFactChecker not? 8 months ago:
Would you then be posting your conclusions? Like, if you’re gonna do that work on some of these posts anyway… may as well share.
- Comment on Why are people downvoting the MediaBiasFactChecker not? 8 months ago:
Bravo for bringing the notes. On a first glance, some of these feel like they require subjectivity (like, do we really believe the political spectrum is 1d?), but I agree I could run the computation myself from this.
- Submitted 10 months ago to [deleted] | 13 comments
- Comment on How do I stop hating children? 1 year ago:
But I think blaming children for the fact that all people are unbearable is… idk, you’ve mistaken a symptom for a problem? Working on the general misanthropy is probably a better start?
- Comment on How do I stop hating children? 1 year ago:
Can I just say that it’s very weird to me that you’re only listing loud things children do… Like, have you ever been around a sleeping child? Do they bother you? Average volume of a child is higher than adults, but only by a factor of 2 or so. And their noises are honest, unlike the adult noises.
- Submitted 1 year ago to videos@lemmy.world | 7 comments
- Comment on What are the most mindblowing fact in mathematics? 1 year ago:
Oh that’s cool - I had heard one or two examples only. Is there some popular writeup of the story from Savant’s view?
- Comment on What are the most mindblowing fact in mathematics? 1 year ago:
An arithmetic miracle:
Let’s define a sequence. We will start with 1 and 1.
To get the next number, square the last, add 1, and divide by the second to last. a(n+1) = ( a(n)^2 +1 )/ a(n-1) So the fourth number is (2*2+1)/1 =5, while the next is (25+1)/2 = 13. The sequence is thus:
1, 1, 2, 5, 13, 34, …
If you keep computing (the numbers get large) you’ll see that every time we get an integer. But every step involves a division! Usually dividing things gives fractions.
This last is called the somos sequence, and it shows up in fairly deep algebra.
- Comment on What are the most mindblowing fact in mathematics? 1 year ago:
I now recall there was a numberphile with exactly that visualisation! It’s a clever visual
- Comment on What are the most mindblowing fact in mathematics? 1 year ago:
For the uninitiated, the monty Hall problem is a good one.
Start with 3 closed doors, and an announcer who knows what’s behind each. The announcer says that behind 2 of the doors is a goat, and behind the third door is
a carstudent debt relief, but doesn’t tell you which door leads to which. They then let’s you pick a door, you get what’s behind the door. Before you open it, they open a different door than your choice and reveals a goat. Then the announcer says you are allowed to change your choice.So should you switch?
The answer turns out to be yes. 2/3rds of the time you are better off switching. But even famous mathematicians didn’t believe it at first.
- Comment on What are the most mindblowing fact in mathematics? 1 year ago:
Note you’ll need the regions to be connected (or allow yourself to color things differently if they are the same ‘country’ but disconnected). I forget if this causes problems for any world map.