If we let p=10*x^2+17*x+3, Maxima properly computes
factor(p)=(2*x+3)*(5*x+1). Its algorithm can be interpreted as finding
factorizations if the field of rational numbers (by Gauss' lemma it suffices
to find factorizations over the integers). However, in the field
Q(sqrt(2)), i.e. all numbers of the form r1+r2*sqrt(2), where r1 and r2 are
rational, another factorization is possible, e.g.
p=(-2+2*sqrt(2))*x-3+3*sqrt(2))*((5+5*sqrt(2))*x+1+sqrt(2)). Is it possible
to get Maxima to find such factorizations? One could do so manually for a
fixed entension field by setting up a set of simultaneous equations to be
solved, but I wonder if there is a quicker, more elegant approach, in which
Maxima even discovers the right extension field.
Thanks, George Leeman