Timeline for A formula for a generator of the multiplicative group of $\mathbb{F}_p$ ?
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 19, 2013 at 20:48 | comment | added | Matthieu Romagny | addendum to the last comment: for a neat statement and proof written for master students, you may have a look at perso.univ-rennes1.fr/matthieu.romagny/agreg/theme/…. (This is in french.) | |
| Feb 19, 2013 at 20:43 | comment | added | Matthieu Romagny | A complete statement is a bit messy to give, but here it is. Let $V$ be the cat. of vector spaces and $I$ the cat. of isom's $E\to F$. Let $S,T:I\to V$ be the source and target functors. Statement: there is no functor $A:V\to I$ such that $S\circ A=Id$ and $T\circ A$ is the functor taking a vector space to its dual. Perdon me for being sloppy about variance considerations with the latter functor, I guess that you see what I mean. | |
| Feb 19, 2013 at 19:31 | comment | added | Joël | Dear Matthieu, can you clarify your example about the non-canonicity of the isomorphism $E \rightarrow E^\ast$. Your source category is the category of vector space of dimension $n$, and your statement is that there is no functor from that category to ... what exactly? | |
| Feb 19, 2013 at 19:14 | comment | added | boumol | Let me add that the first difficulty to address your question is precisely what is the meaning of "formula". I suggest you to read the paper "Formulas for primes" jstor.org/stable/2690261 to realize that although formulas fascinate they are sometimes non interesting. Indeed, this paper is a very funny reading that I recommend to all mathematicians: I never thought about these issues before reading this paper, it opened my mind to some difficulties with the notion of "formula". | |
| Feb 19, 2013 at 15:52 | comment | added | Matthieu Romagny | @quid: that is exactly the point I'd like to clarify. I guess you agree that we often hear that "there is no canonical generator". Among the people that use the word "canonical", some can't give a precise meaning to it and some can. Most of the latter will relate canonicity to some form of functoriality. If some of these people are around reading us, I would like to know if they have such a categorical statement to offer. | |
| Feb 19, 2013 at 14:33 | comment | added | user9072 | I am never quite sure regarding this functorial things but: What should be the categorty? If you take fields of prime order, there are no nonidentity isomorphisms, so just take the smallest as I said (does anything go wrong?). Now if you just take the multiplicative structure, then I'd guess it is the same as asking for a generator of a cyclic group, which I guess is classical. | |
| Feb 19, 2013 at 14:13 | history | edited | Matthieu Romagny | CC BY-SA 3.0 |
added 623 characters in body
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| Feb 19, 2013 at 13:33 | comment | added | Stefan Kohl♦ | Questions for generators of the multiplicative group of $\mathbb{F}_p \cong \mathbb{Z}/p\mathbb{Z}$ tend to be very difficult, see e.g. en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots. | |
| Feb 19, 2013 at 13:30 | answer | added | user9072 | timeline score: 5 | |
| Feb 19, 2013 at 13:28 | answer | added | Felipe Voloch | timeline score: 7 | |
| Feb 19, 2013 at 12:54 | comment | added | user9072 | Please use the existing toplevel-tags (those with two letters prefix) if they exist. (I changed it.) | |
| Feb 19, 2013 at 12:53 | history | edited | user9072 |
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| Feb 19, 2013 at 11:24 | comment | added | tomasz | Related: math.stackexchange.com/questions/124408/… | |
| Feb 19, 2013 at 11:01 | history | asked | Matthieu Romagny | CC BY-SA 3.0 |