Timeline for Elementary Luroth theorem proof?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 28, 2012 at 14:14 | vote | accept | zroslav | ||
| Dec 28, 2012 at 14:14 | answer | added | zroslav | timeline score: 0 | |
| Dec 16, 2012 at 12:46 | answer | added | A_H | timeline score: 5 | |
| Jun 12, 2011 at 15:41 | comment | added | zroslav | @Jim: I have found an elementary proof of Luroth theorem. In the book of A.Schinzel. | |
| Jun 6, 2011 at 18:54 | comment | added | zroslav | @Jim: Ok, but I'm sure that there must be purely elementary proof. | |
| Jun 6, 2011 at 17:38 | comment | added | Jim Humphreys | @zroslav: The theorem just states that any subfield of the big field properly containing the ground field must be a purely transcendental extension (necessarily simple, by comparison of transcendence degrees). This requires some discussion of fields including the distinction between algebraic and transcendental extensions. | |
| Jun 6, 2011 at 16:43 | comment | added | zroslav | @Jim: But when I'm proving that every subalgebra of $k[t]$ is finitely generated I don't need any ring theory. | |
| Jun 6, 2011 at 16:35 | comment | added | zroslav | @Pete: Ritt's theorem on commuting polynomials | |
| Jun 6, 2011 at 15:49 | comment | added | Jim Humphreys | Besides adding a tag, I'd second some of the other comments concerning motivation and usable tools for this purpose. I've never seen a really "elementary" proof of the theorem but absorbed some of Jacobson's viewpoint while a student (as Winter did). Even a super-clever proof must use something from field theory. | |
| Jun 6, 2011 at 15:45 | history | edited | Jim Humphreys |
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| Jun 6, 2011 at 12:59 | comment | added | Emerton | What happens if you take an element of least degree (in the same sense of Charles Matthews's answer below) in the subfield $L$? Is there any chance of showing directly, by some sort of induction on the degree, that such an element generates $L$ as a field? | |
| Jun 6, 2011 at 12:52 | answer | added | Charles Matthews | timeline score: 1 | |
| Jun 6, 2011 at 11:58 | comment | added | Pete L. Clark | I must say that I am very curious about a presentation of Luroth's Theorem for high schoolers which is not motivated by complex analysis or linear algebra. How did you hit upon this topic? What is your angle on it? | |
| Jun 6, 2011 at 9:22 | comment | added | zroslav | @Torsten: Also I do not want to use linear algebra | |
| Jun 6, 2011 at 8:31 | comment | added | Torsten Ekedahl | There is an elementary proof in Winter: The structure of fields, Graduate Texts in Mathematics 16, Springer. It does use a little bit of field theory however but that could possible be whittled away. | |
| Jun 6, 2011 at 7:37 | history | asked | zroslav | CC BY-SA 3.0 |