Timeline for Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 13, 2015 at 6:52 | vote | accept | Pablo | ||
| Jul 13, 2015 at 19:09 | comment | added | David E Speyer | Suppose that $\ell$ and $q$ are odd primes, with $q \equiv 1 \bmod \ell$ and $a \in \mathbb{F}_Q$ is not an $\ell$-th power. Then $x^{\ell^n} - a$ is irreducible for every $n$. In particular, $T^{\ell} - a$ is irreducible infinitely often. See math.stackexchange.com/a/413065/448 , and also other answers there. | |
| Jul 12, 2015 at 8:30 | comment | added | Pablo | @DavidSpeyer are there (nonlinear) cases in which we know an infinite family of irreducible examples? | |
| Jul 11, 2015 at 17:42 | comment | added | David E Speyer | I should probably point also out that this statement is about known cases of the full asymptotic conjecture, not about cases where there are simply known to be infinitely many examples. | |
| Jul 11, 2015 at 11:13 | comment | added | Lior Bary-Soroker | It is important to add that Chris Hall, in his PhD, showed that there are infinitely many $f$ with both $f$ and $f+1$ irreducible (his argument used the the field has at least 4 elements, I think) | |
| Jul 11, 2015 at 7:39 | comment | added | Pablo | Very interesting! What if we replace 'irreducible' by 'squarefree' ? The lemma seems to be still correct, and for squarefree values more is known, as can be seen from m.qjmath.oxfordjournals.org/content/early/2015/07/04/… | |
| Jul 11, 2015 at 2:59 | history | answered | David E Speyer | CC BY-SA 3.0 |