First of all, the polynomial $p(x)=x^{3}-x-4$ must have one real root. Let $\theta$ be one of its roots, L=Q($\theta$), $O$ the integral closure of Z in L.
Then Kummer's theorem tells us then and there that for a prime q not to be nonsplit it is necessary and sufficient that $p(x) \bmod q$ is reducible.
Nevertheless, it in general is pretty hard to determine whether or not it is true, e.g. hard enough for the case q=the biggest prime ever known until now.
So, my question is that is there an efficient way determining the solvability of this kind of polynomials, or if I mistakenly thought of something pretty easy as very difficult, please inform me, thanks.
Edit:An efficient way means it might work for cases such as q=109 since for small enough primes, it suffices to check by hands, such as for 29, (29)=$P_1$P_2$P_3$ where $P_1$=29$O$+($\theta-5)O$,$P_2$=29$O$+($\theta-19)O$, $P_3$=29$O$+($\theta+15)O$, and some similar result for (5) while (5) is not totally split in $O$.
Decomposition of prime ideals
1
$\begingroup$
algebraic-number-theory
-
0Sorr$y$, I will improve it. – 2011-02-14
1 Answers
2
For a particular prime, you can use Cantor-Zassenhaus or another factoring algorithm to factor the polynomial modulo the prime.
-
0I know there is a well-developed theory of ramification in the Galois case, nonetheless, this case is not Galois, and hence I would like to know if there is any result on the direction. – 2011-02-16