Timeline for What is the topology on the set of field orders
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2019 at 19:08 | comment | added | Asaf Karagila♦ | Obligatory SMBC reference. | |
| Oct 30, 2019 at 19:06 | comment | added | Denis Nardin | @AsafKaragila Take it up with whoever thought it was a good idea to call the unique complete ordered field the "real" numbers (not like those fake numbers like $i$)... | |
| Oct 30, 2019 at 16:45 | comment | added | Asaf Karagila♦ | The real spectrum! As opposed to the fake spectrum which appears often in the liberal, left-wing preprint suppositories! | |
| Oct 30, 2019 at 9:56 | history | edited | Denis Nardin | CC BY-SA 4.0 |
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| Oct 30, 2019 at 9:55 | comment | added | YCor | Reference to Harrison topology: Harrison, D. K. Finite and infinite primes for rings and fields. Mem. Amer. Math. Soc. No. 68, 1966, 62 pp. | |
| Oct 30, 2019 at 9:53 | history | edited | Denis Nardin | CC BY-SA 4.0 |
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| Oct 30, 2019 at 9:29 | comment | added | Denis Nardin | @HenrikRüping There is a naturally defined map from $\mathrm{Sper}\,\mathbb{Q}(x)$ to $\mathbb{P}^1(\mathbb{R})$, whose fibers are either $\{\pm 1\}$ or just a point depending if the target is algebraic or not. This can be generalized to a map from $\mathrm{Sper}\,F$ to the $\mathbb{R}$-points of the Zariski-Riemann space of $F$, and that's what you're seeing. The fiber over a point $(v,k(v)\to\mathbb{R})$ is given (noncanonically) by the set of homomorphisms from the valuation group of $v$ to $\{\pm1\}$. | |
| Oct 30, 2019 at 9:25 | comment | added | HenrikRüping | It seems as if at least for the upper example the set of all orders is ordered itself and that there is a canonical quotient map identifying $a^\pm$ such that the quotient space is $[-\infty,\infty]$. This seems really similar to the setting where one takes the standard cantor set and identifies both endpoints of each removed interval and the quotient space is again a unit interval. | |
| Oct 30, 2019 at 8:56 | vote | accept | HenrikRüping | ||
| Oct 30, 2019 at 8:50 | history | edited | Denis Nardin | CC BY-SA 4.0 |
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| Oct 30, 2019 at 8:38 | history | answered | Denis Nardin | CC BY-SA 4.0 |