Timeline for Computation of a minimal polynomial
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 3, 2017 at 19:59 | comment | added | J. M. isn't a mathematician |
Mathematica would also allow you to do things like MinimalPolynomial[Root[#^5 - # + 1 &, 1] - Root[#^5 - # + 1 &, 5], x], to use @Anton's example.
|
|
| Sep 3, 2017 at 17:49 | comment | added | Igor Rivin | @Anton The problem is that you cannot algebraically distinguish one from another. So, some differences $\alpha - \beta$ of roots of your favorite polynomial are equal to zero, so their minimal polynomial is $x.$ The characteristic polynomial of the rational number $\alpha - \beta$ will be thus divisible by powers of $x.$ | |
| Sep 3, 2017 at 16:59 | comment | added | Anton | Thanks, I appreciate that! Though I don't quite understand what do you mean by "characteristic polynomial will be reducible". Say, I know that if a and b are two distinct roots of x^5-x+1, then the minimal polynomial of a-b is x^20-10*x^16-...+5000x^2+2869, a.k.a. the characteristic polynomial of a companion matrix of a-b, is irreducible. What's the problem here? | |
| Sep 3, 2017 at 16:43 | history | edited | Igor Rivin | CC BY-SA 3.0 |
added example
|
| Sep 3, 2017 at 16:40 | comment | added | Igor Rivin | @Anton In that case, the characteristic polynomial will be reducible (because the roots of the characteristic polynomials will also include things like $2\alpha.$). I will add mathematica code. | |
| Sep 3, 2017 at 16:34 | vote | accept | Anton | ||
| Sep 3, 2017 at 16:28 | history | edited | Igor Rivin | CC BY-SA 3.0 |
fixed paren
|
| Sep 3, 2017 at 16:26 | comment | added | Anton | Thank you, Igor! I think this is precisely what I was looking for. Can you give an example of how this works w.r.t. two distinct roots of x^5-x+1? I don't quite understand how this works when two distinct algebraic numbers have the same companion matrix. | |
| Sep 3, 2017 at 16:13 | history | answered | Igor Rivin | CC BY-SA 3.0 |