Timeline for Basis for the Algebraic numbers over the rationals
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 3, 2010 at 14:25 | vote | accept | mathahada | ||
| Dec 2, 2010 at 12:58 | comment | added | Joel David Hamkins | Mathahada, I appreciate that you accepted my answer, but it would seem more appropriate for you to accept Russell Miller's answer, because he gave the solution to the version of your question that I proposed. (And also Jared Weinstein suggested a similar solution in comments.) | |
| Dec 2, 2010 at 12:47 | vote | accept | mathahada | ||
| Dec 3, 2010 at 14:25 | |||||
| Dec 2, 2010 at 3:30 | answer | added | Russell Miller | timeline score: 13 | |
| Dec 1, 2010 at 2:38 | answer | added | Joel David Hamkins | timeline score: 6 | |
| Nov 30, 2010 at 18:05 | comment | added | Kevin Buzzard | Sure! Just enumerate the rationals using the usual "spiral" argument, enumerate the polynomials with rational coefficients using lexicographic ordering, enumerate the complex roots of each of them using some suitable partial ordering on the complexes (e.g. $z_1<z_2$ iff $z_1$ has a smaller argument than $z_2$, or they have the same argument but $|z_1|<|z_2|$) and now you've enumerated the algebraic numbers and you can just now construct a basis by going through each one adding it in if it's not a linear combination of anything we've seen before. Completely explicit! ;-) | |
| Nov 30, 2010 at 16:49 | comment | added | Will Jagy | I would bet against it. I once asked T. Y. Lam about preferred forms for compass and straightedge "constructible numbers," those being the smallest field extension of the rationals closed under square roots of positive elements. He was quite firm that there is no canonical form for these numbers. $$ $$ Meanwhile, you are asking for far more intricate fields. Perhaps it depends on the word "explicit." | |
| Nov 30, 2010 at 15:55 | history | asked | mathahada | CC BY-SA 2.5 |