Timeline for Transformations of integer polynomials under combinations of their roots
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 29, 2011 at 15:43 | vote | accept | Niel de Beaudrap | ||
| Apr 15, 2010 at 23:52 | answer | added | Gerry Myerson | timeline score: 3 | |
| Apr 15, 2010 at 15:33 | answer | added | Qiaochu Yuan | timeline score: 0 | |
| Apr 15, 2010 at 11:56 | answer | added | Jacques Carette | timeline score: 1 | |
| Apr 15, 2010 at 11:56 | comment | added | Niel de Beaudrap | I think I understand why you were addressing Q2 as opposed to Q1, and I can answer. While there exist reducible polynomials whose principal roots are u+v, uv, etc. only if there exist irreducible ones, I allowed for the possibility that sometimes the reducible ones might be easier to find than the irreducible ones. (That is, lifting the requirement that the polynomial be irreducible can only make the problem of constructing a suitable polynomial easier.) | |
| Apr 15, 2010 at 9:06 | comment | added | Niel de Beaudrap | I am aware of this concept; indeed I am implicitly invoking it for the relationship between the negation and the inverse of the golden ratio, as well as my comments about differences of principal roots after my questions. You make a good point about products of "large", negative, non-principal roots. (I'm not sure how it relates to question 2 in particular, as opposed to question 1.) I will make a revision to the question, in light of this problem | |
| Apr 15, 2010 at 9:05 | history | edited | Niel de Beaudrap | CC BY-SA 2.5 |
Revised question in response to comments
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| Apr 15, 2010 at 8:29 | comment | added | Kevin Buzzard | Regarding Q2: Are you aware of the notion of the conjugates of an algebraic number? If alpha is algebraic then any polynomial f in Z[x] with alpha as a root will have all the conjugates of alpha as roots as well. Hence your 2nd question seems to be based on a misapprehension. If u has a conjugate u' with u'<0<u and if v has a conjugate v' with v'<0<v then u'v' will be a conjugate of uv and it might be bigger than uv, and in this case uv can never be the principal root of any polynomial at all. An explicit example would be u=sqrt(2)-1 and v=sqrt(3)-1. | |
| Apr 15, 2010 at 7:40 | history | asked | Niel de Beaudrap | CC BY-SA 2.5 |