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revised Cases where the number field case and the function field (with positive characteristic) are different
added 358 characters in body
2d
awarded  Nice Answer
2d
comment Cases where the number field case and the function field (with positive characteristic) are different
@StanleyYaoXiao, I worked out some examples myself now, and included one of them in my edit to the answer.
2d
revised Cases where the number field case and the function field (with positive characteristic) are different
added 801 characters in body
Jul
24
awarded  Good Question
Jul
20
comment Cases where the number field case and the function field (with positive characteristic) are different
@StanleyYaoXiao, if you want to run some numerical tests in $\mathbf F_q[x]$ (say $q = 3$ or $5$) for the Carmichael case of $f = \pi_1\pi_2$ with factors of equal degree, to see how close it comes to 50%, let me know how things turn out.
Jul
19
awarded  Necromancer
Jul
19
comment Magic trick based on deep mathematics
Sure, and I wouldn't say that either. Read again what I wrote in the last paragraph of my answer: when doing the trick I tell people to pick a number from 1 to 7. They've never picked 1.
Jul
18
answered Cases where the number field case and the function field (with positive characteristic) are different
Jul
18
comment Magic trick based on deep mathematics
I wouldn't because that sounds artificial.
Jul
13
awarded  Nice Answer
Jul
10
comment Genus of $k(T)$ is $0$ without using Riemann-Roch
What is your definition of the genus of a function field over $k$?
Jul
7
awarded  Good Answer
Jul
6
comment About Abhyankar's conjecture
@WillSawin, you have an (autocorrect?) error in the type of covers described in your comment.
Jul
5
awarded  Necromancer
Jul
4
comment Stability of the Solar System
Although it's been about 10 years since Pluto was demoted from the list of planets, it still looks strange to me to see "8 planets" instead of 9. Perhaps billions of years from now there will be 15 planets after the opposing astronomers make another definition of what a planet is that lets in Pluto and its cousins. A relevant question is whether the definition of a planet will remain stable.
Jul
3
comment Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?
Yes. Collect the terms with $f$ of degree $n$ together. The coefficient of $1/q^{ns}$ is $\sum_{\deg f = n} \chi(f)c^n$, with the sum running over monic $f$. For $n = 0$ the sum is $1$. That the sum vanishes when $n \geq \deg g$ is Proposition 4.3 in Rosen's Number Theory in Function Fields. So when $g$ is linear the $L$-series is just the constant term, which is $1$.
Jul
3
comment History of Geometric Analogies in Number Theory
Related question (follow the links to papers by Roquette): mathoverflow.net/questions/16343/…
Jun
30
comment Graduate program applications that require questionnaires and other non-letter material
@Kimball, if someone already has a stackexchange account I don't think it's a big burden to pick up another one, particularly that one. I can imagine there might be some reluctance to set up an account for the "worldbuilding" stackexchange site, but I've seen mathematicians answering questions on the academia site.
Jun
26
revised Does the proof of Picard's theorem become simpler by increasing the number of points that are not attained?
edited title