If anyone knows where to find an equivalent concept, definitely let me know! It’s hard to prove that a concept has not been investigated before, so I’ll just give my reasoning on why I think it’s unlikely:
I believe that the name “n-cohesive” is untethered to the literature, even if the underlying concept exists, since it was specifically told to change the name to something new.
The original name, “n-smooth,” appears in number theory, but the number theory concept is genuinely unrelated. The word “smooth” is heavily overloaded in algebra, but generally points towards a “main” concept called “smoothness of a morphism,” and smoothness of a morphism is incapable of caring about which power of a prime divides something. So it is at least unrelated to the main uses of the word “smooth.”
When you start doing math research, you have to develop something resembling an “originality” radar. Mine is far from perfect, and only really works well in a narrow, narrow corner of ring theory. But for what it’s worth, it’s set off by the concept. It’s hard to communicate this sort of intuition, but subjectively to me, it contributes to my belief.
Cultural exposure is part of it. If you tried to pass off a made-up geometry word to me, it would be pretty easy for you to succeed (I’m not a geometer), but, at least I would like to believe, my “fake-detector” for ring theory is a bit more calibrated, and not only does this not trip the “I’ve heard of this alarm,” it doesn’t even really sound similar to other points in the cluster “unit group stuff” out in conceptspace. I know that generators of unit groups are important, and orders of generators are probably important by proxy, but why care about order related to the residual characteristic? It sets off my flag as “weird” and “I’ve not heard of this sort of thing before,” which triggers a subjective belief “I don’t think people have done something like this.”
In general, concepts in ring theory about a prime dividing (or not dividing) the order of something exist, and concepts where the prime has something to do with the characteristic are certainly important, but almost always when the characteristic is prime. People just don’t really talk about non-prime characteristic that much. And the same power for different primes? Even if those primes divide the characteristic with different multiplicity? That’s just odd.
Stepping out of what the AI can do, and what a human might realize: in Z, the unit group of a quotient Z/n has order equal to Euler’s totient function of n, which not only involves the prime factors of n, but those prime factors minus 1. n-cohesiveness then asks about powers of a prime dividing the totient, which implicitly requires you to understand the prime factors of various primes minus 1, which is generally very hard. I am genuinely not sure that I could classify 5-cohesive ideals of Z. It doesn’t feel like a problem I’ve seen people try to work on before. And that’s just Z, forget Z[√−19] or some cyclotomic extension.
If anyone knows where to find an equivalent concept, definitely let me know! It’s hard to prove that a concept has not been investigated before, so I’ll just give my reasoning on why I think it’s unlikely:
I believe that the name “n-cohesive” is untethered to the literature, even if the underlying concept exists, since it was specifically told to change the name to something new.
The original name, “n-smooth,” appears in number theory, but the number theory concept is genuinely unrelated. The word “smooth” is heavily overloaded in algebra, but generally points towards a “main” concept called “smoothness of a morphism,” and smoothness of a morphism is incapable of caring about which power of a prime divides something. So it is at least unrelated to the main uses of the word “smooth.”
When you start doing math research, you have to develop something resembling an “originality” radar. Mine is far from perfect, and only really works well in a narrow, narrow corner of ring theory. But for what it’s worth, it’s set off by the concept. It’s hard to communicate this sort of intuition, but subjectively to me, it contributes to my belief.
Cultural exposure is part of it. If you tried to pass off a made-up geometry word to me, it would be pretty easy for you to succeed (I’m not a geometer), but, at least I would like to believe, my “fake-detector” for ring theory is a bit more calibrated, and not only does this not trip the “I’ve heard of this alarm,” it doesn’t even really sound similar to other points in the cluster “unit group stuff” out in conceptspace. I know that generators of unit groups are important, and orders of generators are probably important by proxy, but why care about order related to the residual characteristic? It sets off my flag as “weird” and “I’ve not heard of this sort of thing before,” which triggers a subjective belief “I don’t think people have done something like this.”
In general, concepts in ring theory about a prime dividing (or not dividing) the order of something exist, and concepts where the prime has something to do with the characteristic are certainly important, but almost always when the characteristic is prime. People just don’t really talk about non-prime characteristic that much. And the same power for different primes? Even if those primes divide the characteristic with different multiplicity? That’s just odd.
Stepping out of what the AI can do, and what a human might realize: in Z, the unit group of a quotient Z/n has order equal to Euler’s totient function of n, which not only involves the prime factors of n, but those prime factors minus 1. n-cohesiveness then asks about powers of a prime dividing the totient, which implicitly requires you to understand the prime factors of various primes minus 1, which is generally very hard. I am genuinely not sure that I could classify 5-cohesive ideals of Z. It doesn’t feel like a problem I’ve seen people try to work on before. And that’s just Z, forget Z[√−19] or some cyclotomic extension.