Timeline for What is the definition of "canonical"?
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6 events
| when toggle format | what | by | license | comment | |
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| Nov 13, 2013 at 23:47 | comment | added | Simon Henry | @Mathieu. As far as I am concern, this fact "there is no canonical generator for the multiplicative group of the field with p elements?" Is false. There is canonical generator : the one which is the smallest if I identify $F_p$ with {0,1,..p-1}. But I guess it depends on your interpretation of "canonical" | |
| Feb 17, 2013 at 13:30 | comment | added | Matthieu Romagny | If we are considering the category of fields with $p$ elements, then the canonical generator $c$ in Bourbaki's sense works also in the functorial sense since an isomorphism between two fields with $p$ elements takes $c$ to $c$. (Of course there is only one isomorphism, and it is used to identify $c$ everywhere...) Maybe a more satisfying non-existence statement in the functorial sense of canonicity is that cyclic groups of order $n$ do not have a canonical generator if $n\ge 3$. | |
| Feb 17, 2013 at 13:02 | comment | added | Matthieu Romagny | OK, thanks! (Note: for $p=3$ also, the generator is canonical.) | |
| Feb 16, 2013 at 23:48 | comment | added | ACL | Well (except when $p=2$), for every generator of the multiplicative group $\mathbf F_p^\times$, there is an automorphism of this group which moves the generator. But there is a canonical generator in the former sense: I would say that the one with smallest positive representative in $\mathbf Z$ qualifies. | |
| Feb 16, 2013 at 19:50 | comment | added | Matthieu Romagny | Dear Antoine, how do you apply this categorical interpretation to prove the following fact: there is no canonical generator for the multiplicative group of the field with $p$ elements? | |
| Feb 16, 2013 at 15:02 | history | answered | ACL | CC BY-SA 3.0 |