I'd like to factor efficiently polynomials over rings (more particularly the rings of the form IntegerModRing(n), other rings don't interest me right now). I've noticed that you can factor a polynomial differently when considering FIELDS, so that something like
factor(x^2-2, QQ[]) factor(x^2-2, RR[])
will output the different expected results. But what about
x = var('x') factor(x^5-x, IntegerModRing(25)['x'])?
What is actually outputted right now is
(x-1)(x+1)(x^2+1)*x
but the true factorization is x(x-1)(x+1)(x-7)(x+7)
(I computed it by hand, Sage couldn't do it.) Going for small numbers like 25 is easy but when I go for large numbers I get nasty codes if I try to compute that myself. Isn't there anyway to make the factor command factor those polynomials or another way to do it? Any suggestion is welcome.
I am new to Sage and I am beginning to love its features since I begin to study number theory and Sage has plenty of options for that purpose, but I must admit they are quite hard to understand since I am also new to Python, PARI, Magma... (not to programming though!) explanations in detail would be deeply appreciated.
