Here are the two different factorization results of $\mathfrak{p} = 5$ in two different number fields $F = \mathbb{Q}[x]/(x^3 + 10x - 12)$ and $G = \mathbb{Q}[x]/(x^3 - 6x + 41)$.
Type ?17 for how to get moral (and possibly technical) support.
parisize = 8000000, primelimit = 500000
? F = nfinit(x^3 + 10*x - 12)
? G = nfinit(x^3 - 6*x - 41)
The prime ideal $\mathfrak{p} = 5$ factors in two different ways depending on the number field:
? idealfactor(F,5)
%3 =
[[5, [2, 1, 0]~, 1, 1, [-2, 24, 0; -2, -6, 10; 2, -4, 0]] 1]
[ [5, [-2, -2, 2]~, 1, 2, [2, -6, 6; 1, 2, -2; 0, 2, 2]] 1]
? idealfactor(G,5)
%4 =
[[5, [-2, 1, 0]~, 1, 1, [-2, -34, -8; -2, -4, -8; -2, -4, 2]] 1]
[ [5, [-2, -2, -2]~, 1, 2, [-2, 5, 12; 1, -1, 0; 0, 3, -3]] 1]
As a language check, let's try noticing that $5 = \mathfrak{p}_1\mathfrak{p}_2$ with $\mathfrak{p}_1 +\mathfrak{p}_2 = (1) = \mathcal{O}_F$ (the two factors are relatively prime).
? p1 = idealprimedec(F,5)[1]
%7 = [5, [2, 1, 0]~, 1, 1, [-2, 24, 0; -2, -6, 10; 2, -4, 0]]
? p2 = idealprimedec(F,5)[2]
%8 = [5, [-2, -2, 2]~, 1, 2, [2, -6, 6; 1, 2, -2; 0, 2, 2]]
? idealadd(F,p1,p2)
%9 =
[1 0 0]
[0 1 0]
[0 0 1]
From the pari-gp documentation, and from Wikipedia it looks like the terms maximal order and ring of integers are the same (as terms in ring theory. $(5) = (5,2 + a)(5,-2 -2a + 2a^2)$ as ideals in $\mathcal{O}_F$. Could this be written as a factorization of integer matrices?