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14h

revised 
Cases where the number field case and the function field (with positive characteristic) are different
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awarded  Nice Answer 
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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 RiemannRoch
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 nonletter 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 