Timeline for How to decide if an algebraic number is a root of a given polynomial?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 16, 2021 at 7:50 | comment | added | Jára Cimrman | @GerryMyerson When it comes to my comment about reducibility of cyclotomic polynomials over some number fields: the fact that the factorization of $x^4 + 4$ is "finer" than the one of $x^4 + 1$ can be seen as a consequence of the fact that the cyclotomic polynomial $x^4 + 1$ is reducible over $\mathbb{Q}(\sqrt{2})$. Hence my skepticism to the claim that the minimal polynomial can be easily expressed in terms of cyclotomic polynomials. | |
| Aug 16, 2021 at 7:16 | comment | added | Jára Cimrman | @GerryMyerson Your link gives a necessary and sufficient condition for irreducibility of $x^r - b$. For instance, both $x^9 - 8$ and $x^8 - 16$ are reducible: $x^9 - 8 = (x^3-2)(x^6+2x^3+4)$ and $x^8 - 16 = (x^4-4)(x^4+4)$. In this case it can be "detected", to which of the factors the root $\alpha$ corresponds. If $\alpha$ is a root of the first factor, then one can indeed proceed recursively. But what about the remaining case? Above, $x^6+2x^3+4$ is irreducible, but $x^4+4$ is reducible. Can the remaining factor be of the form other than $x^n\pm a$ and still be reducible? | |
| Aug 16, 2021 at 3:35 | comment | added | Gerry Myerson | It comes down to a question of polynomials of the form $x^r-b$ for $b$ rational. The factorization of such polynomials over the rationals is well-understood. In particular, exceptional cases are dealt with at math.stackexchange.com/questions/133581/… Since $p$ is assumed to have rational coefficients, I don't see how reducibility over fields other than the rationals can be a problem. | |
| Aug 15, 2021 at 16:44 | comment | added | Jára Cimrman | @NeilStrickland Thank you! Could you, please, elaborate on how the minimal polynomial looks like, or how can it be computed? This is the problem I have been originally motivated by. :) (To me, as a layman, it seems that potential reducibility of cyclotomic polynomials over some number fields may complicate things.) | |
| Aug 15, 2021 at 16:41 | comment | added | Neil Strickland | For $\alpha$ of the indicated type it is straightforward to write down the minimal polynomial $f(x)\in\mathbb{Z}[x]$ in terms of cyclotomic polynomials, then you just have to divide $p(x)$ by $f(x)$ and see if there is a remainder, which is easily done in polynomial time. | |
| Aug 15, 2021 at 16:29 | history | asked | Jára Cimrman | CC BY-SA 4.0 |