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8h

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Stability of 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. 
17h

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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 $1$. 
1d

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History of Geometric Analogies in Number Theory
Related question (follow the links to papers by Roquette): mathoverflow.net/questions/16343/… 
Jun 30 
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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 
Jun 26 
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Number of $\mathbb F_p$ points constant mod $p$?
@JasonStarr, whoops. Got it. 
Jun 25 
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Number of $\mathbb F_p$ points constant mod $p$?
Can you work out a complete formula for $\#X({\mathbf F}_p)$ as $p$ varies, e.g., compute the zetafunction of $X_{/{\mathbf F}_p}$? Perhaps the counting formula for $\#X({\mathbf F}_p)$ is a universal polynomial in $p$ (like Hall polynomials). 
Jun 25 
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Number of $\mathbb F_p$ points constant mod $p$?
@JasonStarr, is $\mathbf F_q$ a potentially general finite field and not one of prime order? If so, "modulo $q$" is not what you mean (e.g., $q = 8$). 
Jun 23 
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What are applications of commutativity theorems for rings?
@PaceNielsen, thanks. And have you ever seen his commutativity theorem used to prove some type of ring is commutative that is not obviously so a priori, besides finite division rings or rings that are close to boolean rings? 
Jun 23 
awarded  Nice Answer 
Jun 19 
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Which topological properties are preserved under taking box products?
That's a better version. 
Jun 19 
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Which topological properties are preserved under taking box products?
The notation $\models$ is not widely known among people who haven't studied mathematical logic. (I had to look at the code of your question, as if I were to edit it, to know how to typeset it.) Would you consider rewriting the last paragraph so it is understandable to those without familiarity with that notation by writing the question in plain English? 
Jun 17 
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Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?
To see the equivalence that Noam mentions, see Theorem 3.3 (and the two lemmas preceding it) in math.uconn.edu/~kconrad/articles/eulerprod.pdf, setting $d = 1$ and $\alpha_{p,1} = \chi(p)$. Note that for infinite products like an Euler product, the term "converges" means "converges and is not $0$." Since nobody has ever proved the $L$function of a nontrivial Dirichlet character has no zeros in any vertical strip $1\varepsilon < {\rm Re}(s) < 1$, it is basically hopeless at present to expect anyone to prove the Euler product converges at any $s$ with ${\rm Re}(s) < 1$. 
Jun 14 
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root of unity in a local field
Try Serre's "Local Fields." There is a section in it on cyclotomic extensions of $\mathbf Q_p$. Or better yet, just try working this stuff out for yourself. It is an instructive experience. 
Jun 13 
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root of unity in a local field
Have you ever read a book on local fields? 
Jun 13 
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root of unity in a local field
The primeto$p$ part of $n$ divides $q1$, where $p$ is the residue field characteristic. 
Jun 13 
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root of unity in a local field
Try $\mathbf Q_2$ and $n=2$. If you take $n$ to be relatively prime to $q$ then your guess is right. 
Jun 13 
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A new generalisation of Fermat's little theorem?
@PierreDenis, equation (12) in section 3 is the main point for you. It says $\sum_{dm} \mu(m/d)a^d \equiv 0 \bmod m$ for any $a \in \mathbf Z$ (of course it suffices to check only for $1 \leq a \leq m1$). He notes Gauss proved it for prime $a$ and other mathematicians proved it in general by several mathematicians in the 1880s. 
Jun 13 
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A new generalisation of Fermat's little theorem?
This is not new. Google "necklace polynomials" and consider the fact that they are integralvalued. 
Jun 12 
awarded  Enlightened 