Timeline for Analogy between Integers and Permutations
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 21, 2014 at 0:27 | comment | added | Qiaochu Yuan | I"m claiming that there isn't anything being permuted if you only look at $\mathbb{Z}$ (obviously the logarithms of primes are not cycle lengths, e.g. they aren't commensurate), but also that there's no reason to restrict your attention to $\mathbb{Z}$. I'm reasonably confident that you get the same kind of statistics for any Dedekind domain $D$ such that $D/P$ is always finite for all prime ideals $P$. | |
| May 21, 2014 at 0:23 | comment | added | john mangual | your answer is good, but I am not asking about $\mathbb{F}_q[x]$ I am asking about $\mathbb{Z}$. | |
| May 21, 2014 at 0:19 | comment | added | Qiaochu Yuan | I find that comment strange. If you're willing to take an analogy between prime factorization and cycle decomposition seriously I don't see how it could hurt to know that it factors through an analogy to factorization of polynomials over finite fields, which on the one hand is part of a well-established analogy between number fields and function fields (which requires no knowledge of etale cohomology to appreciate) and where on the other hand I can explicitly point to a permutation that determines the factorization. | |
| May 21, 2014 at 0:18 | comment | added | john mangual | I am not asking about function fields, I am asking about number fields and $\mathbb{Z}$ | |
| May 21, 2014 at 0:16 | comment | added | Qiaochu Yuan | What does that comment have to do with my answer? | |
| May 21, 2014 at 0:14 | comment | added | john mangual | I think I understand the Frobenius map really well $x \mapsto x^q$ but I certainly do not get Étale cohomology. | |
| May 21, 2014 at 0:11 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
added 87 characters in body
|
| May 20, 2014 at 23:58 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |